Remarks regarding some Algebras of Logic

1 Introduction

Algebras of Logic are explicit algebraic systems that show the basic mathematical structure of Logic. These algebras deal with some algebraic structures, often bounded lattices, considered as models of certain logics, including logic as a domain of order theory. There are well known their importance and applications in social life to advance useful concepts, as for example computer algebra, also called symbolic computation or algebraic computation, which it is a scientific area that refers to the study and development of algorithms or/and software used for computing mathematical expressions and manipulating other mathematical objects, as well as in the study of truth values from social analyses. In the following, we will exemplify some of the applications of Algebras of Logic.

Basic Logic (BL, for short) was introduced by Hájek in [2] to formalize the many-valued semantics induced on the real interval [0, 1] by a continuous t-norm. Basic Logic generalizes the three most used logics in the theory of fuzzy sets: Łukasiewicz logic, Gődel logic and Product logic. BL-algebras are Lindenbaum-Tarski algebras for Basic Logic.

In [1], Di Nolla and Lettieri analyze the structure of finite BL-algebras. Starting from some of these results, in this paper we will study and find conditions such that a commutative unitary ring R, with a finite given number of ideals, to have or not on its algebra of ideals, Id (R) , a BL-structure. For example, to obtain rings with three ideals such that Id (R) is a BL-algebra which is not an MV-algebra, we must search among non-unitary rings or among non-commutative rings or both, or to find such an algebra as a subset or as a subalgebra of an infinite BL-algebra (see Theorem 3.1, Theorem 3.6, Theorem 3.9, Remark 3.10).

Moreover, we present properties of finite rings or rings with a finite number of ideals and their connections with BL-rings. We give some examples in this purpose and we present in Conclusions directions of research to find classifications of such rings.

In this paper, all considered rings are commutative and unitary rings.

2 Preliminaries

Let R be a commutative unitary ring. The set Id (R) denotes the set of all ideals of the ring R. Let I, J ∈ Id (R). The following sets are also ideals in R :

I + J = < I ∪ J > = { i + j , i ∈ I , j ∈ J } , I ⊗ J = { ∑ n k = 1 i k j k , i k ∈ I , j k ∈ J } , ( I : J ) = { x ∈ R , x · J ⊆ I } , Ann ( I ) = ( 0 : I ) , where 0 = < 0 > , called sum, product, quotient and annihilator of the ideal I.

Proposition 2.1.

1) If (m, n) = 1, then ℤ m [ X ] × ℤ n [ X ] ≃ ( ℤ m × ℤ n ) [ X ] = ℤ mn [ X ] .

2) ℤ [ X ] ( n ) ≃ ( ℤ / ( n ) ) [ X ] .

3) If I = I₁ × I₂ is an ideal in the ring R [X] × R [X], then ( R [ X ] × R [ X ] ) / I ≃ R [ X ] I 1 × R [ X ] I 2 .

Proof. It is obvioulsy, by a direct computation. □

Definition 2.2. ([3]) A (commutative) residuated lattice is an algebra (L, ∧ , ∨ , ⊙ , → , 0, 1) such that:

(1)   (L, ∧ , ∨ , 0, 1) is a bounded lattice;

(2)    (L, ⊙ , 1) is a commutative ordered monoid;

(3)  z ≤ x → y iff x ⊙ z ≤ y, for all x, y, z ∈ L .

The property (3) is calledresiduation, where ≤ is the partial order of the lattice   (L, ∧ , ∨ , 0, 1) .

In a residuated lattice is defined an additional operation: for x ∈ L, we denote x^∗ = x → 0 .

If we preserve these notations, for a commutative and unitary ring we have that (Id (R) , ∩ , + , ⊗ → , 0 = {0} , 1 = R) is a residuated lattice in which the order relation is ⊆, I → J = (J : I) and I ⊙ J = I ⊗ J, for every I, J ∈ Id (R), see [4]

In a residuated lattice (L, ∧ , ∨ , ⊙ , → , 0, 1) we consider the following identities: ( prel ) ( x → y ) ∨ ( y → x ) = 1 ( prelinearity ) ; ( div ) x ⊙ ( x → y ) = x ∧ y ( divisibility ) . Definition 2.3. ([5]) 1) A residuated lattice L is called a BL-algebra if L verifies (prel) and (div) conditions.

2) A BL-chain is a totally ordered BL-algebra, i.e., a BL-algebra such that its lattice order is total.

Definition 2.4. ([6])An MV-algebra is an algebra (L, ⊕ , ^∗, 0) satisfying the following axioms:

(1) (L, ⊕ , 0) is an abelian monoid;

(2) (x^∗) ^∗ = x ;

(3) x⊕ 0^∗ = 0^∗ ;

(4) (x^∗ ⊕ y) ^∗ ⊕ y = (y^∗ ⊕ x) ^∗ ⊕ x, for all x, y ∈ L .

Remark 2.5. If in a BL- algebra L we have x^∗∗ = x, for every x ∈ L, and, for x, y ∈ L, we denote x ⊕ y = ( x ∗ ⊙ y ∗ ) ∗ , then we obtain an MV-algebra structure (L, ⊕ , ^∗, 0) . Conversely, if (L, ⊕ , ^∗, 0) is an MV-algebra, then (L, ∧ , ∨ , ⊙ , → , 0, 1) becomes a BL-algebra, in which for x, y ∈ L we have: x ⊙ y = ( x ∗ ⊕ y ∗ ) ∗ , x → y = x ∗ ⊕ y , 1 = 0 ∗ , x ∨ y = ( x → y ) → y = ( y → x ) → x and x ∧ y = ( x ∗ ∨ y ∗ ) ∗ , see [5].

We recall that in [1], Di Nolla and Lettieri analyze the structure of finite BL-algebras. They introduced the concept of BL-comets, a class of finite BL-algebras which can be seen as a generalization of finite BL-chains. Using BL-comets, any finite BL-algebra can be represent as a direct product of BL-comets.

Definition 2.6. 1) ([1], Definition 21) Let L be a BL-algebra. An element x ∈ L is called idempotent if x ⊙ x = x.

2) Let L be a finite BL-algebra and I ( L ) the set of idempotent elements in L. For x ∈ I ( L ) , we denote C ( x ) = { y ∈ I ( L ) such that x and y are comparable}. We defined the set D ( L ) ⊆ I ( L ) as follows:

x ∈ D ( L ) if and only if

i) C ( x ) = I ( L ) ;

ii) The set { y ∈ I ( L ) , y ≤ x} is a chain.

It is clear that D ( L ) ≠ ∅ since 0 ∈ D ( L ) .

A finite BL-algebra L is called a BL-comet if max D ( L ) ≠ 0 .

We recall that in a BL-comet L, the element max D ( L ) is called the pivot of L and it is denoted by pivot (L) .

Proposition 2.7. ([1], Proposition 26) Let L be a finite BL-algebra. The following assertions are equivalent:

(i) L is a BL-comet and pivot (L) =1 ;

(ii)L is a BL-chain.

Using the characterization of Boolean elements in BL-algebras, we establish the connections between BL-comets and BL(MV)-chains.

Proposition 2.8. Let L be a BL-comet. Then L is a BL-chain iff pivot (L) ^∗∗ = pivot (L) .

Proof. If L is a BL-chain, using Proposition 1.10, pivot (L) =1 and obviously, 1^∗∗ = 1 .

Conversely, if we suppose that pivot (L) ^∗∗ = pivot (L) , using the characterizations of Boolean elements in BL-algebras, see [7, Proposition 3.3], it is clear that pivot (L) is a boolean element in L, so, pivot (L) ∨ pivot (L) ^∗ = 1 . We deduce that pivot (L) ^∗ is also a boolean element, so, an idempotent element in L . Then pivot (L) ^∗ ∈ C (pivot (x)). We conclude that pivot (L) and pivot (L) ^∗ are comparable.

If pivot (L) ≤ pivot (L) ^∗, then 1 = pivot (L) ∨ pivot (L) ^∗ = pivot (L) ^∗, so, pivot (L) =0, a contradiction.

If pivot (L) ^∗ ≤ pivot (L) , then 1 = pivot (L) ∨ pivot (L) ^∗ = pivot (L) . Using Proposition 1.10, L is a BL-chain. □

From the above proposition, we deduce the following result:

Proposition 2.9. Let L be a finite MV-algebra. The following assertions are equivalent:

(i) L is a BL-comet;

(ii)L is an MV-chain.

3 Some remarks regarding commutative unitary rings

Definition 3.1. ([8]) Let R be a commutative unitary ring.

1) The ideal M of the ring R is maximal if it is maximal, with respect of the set inclusion, amongst all proper ideals of the ring R. That means, there are no other ideals different from R contained M. The ideal J of the ring R is a minimal ideal if it is a nonzero ideal which contains no other nonzero ideals.

2) A commutative local ring R is a ring with a unique maximal ideal.

3) Let P ¬ = R be an ideal in the ring R. Let a, b ∈ R such that ab ∈ P. If we have  a ∈ P or b ∈ P, therefore P is called a prime ideal of R.

Remark 3.2. Let R be a commutative unitary ring.

1) The ideal M of the ring R is maximal if and only if R/M is a field;

2) The ideal P of the ring R is prime if and only if R/P is an integral domain.

From here, we have that a maximal ideal is a prime ideal.

For other details and properties, the reader is referred to [8].

Definition 3.3. i) [9] A commutative ring R is called a multiplication ring if for every ideals I₁, I₂ of R with I₁ ⊆ I₂, there is an ideal I₃ of R such that I₁ = I₂ ⊗ I₃.

ii) A commutative ring R is called Noetherian ring if the condition of ascending chain is satisfied, that means every increasing sequence of ideals I₁ ⊆ I₂ ⊆ . . . ⊆ I _r  ⊆ . . . is stationary, therefore there is q such that I _q  = I_q+1 = . . ..

iii) A commutative ring R is called Artinian ring if the condition of descending chain is satisfied, that means every decreasing sequence of ideals I₁ ⊇ I₂ ⊇ . . . ⊇ I _r  ⊇ . . . is stationary, therefore there is q such that I _q  = I_q+1 = . . ..

Remark 3.4.

i) ([10], Lemma 3.5) Let R = ∏ j ∈ J R i be a direct product of rings. R is a multiplication ring if and only if R _j is a multiplication ring for all j ∈ J.

ii) ([10], Lemma 3.6) Let R be a multiplication ring and I be an ideal of R. Therefore,  the quotient ring R/I is a multiplication ring.

Remark 3.5. ([11]. Corollary 6.1) Let R be a ring. The following conditions are equivalent:

i) R [X] is a multiplication ring;

ii) R is a finite direct product of fields.

Remark 3.6. ([12] and [8])

1) If R is a Noetherian ring, therefore the polynomial ring R [X] is Noetherian and the quotient ring R/I is also a Noetherian ring, for I an ideal of R.

2) Any field and any principal ideal ring is a Noetherian ring.

3) Every ideal of the Noetherian ring R is finitely generated.

4) An integral domain R is Artinian ring if and only if R is a field.

5) The ring K [X]/(X ^t ) is an Artinian ring, for K a field and t a positive integer.

6) A commutative Noetherian ring R is Artinian if and only if R is a product of local rings.

7) In an Artinian ring every prime ideal is maximal.

8) An Artinian ring is a finite direct product of Artinian local rings.

Proposition 3.7. Let R be a commutative unitary ring which has exactly three ideals {0} , I, R. Therefore, we have I² = {0}.

Proof. It is clear that I is the only minimal ideal in R. Since I² ⊆ I, we have I² = {0} or I² = I. In the same time, I is the only maximal ideal of R and each element from I is a non-unit. Suppose that I² = I. Let x ∈ I such that x ≠ 0. It is clear that I =< x > and x² ≠ 0. We consider the ideal A = {z ∈ R / xz = 0}.We prove that A ≠ {0}. Since x² ≠ 0, therefore there is an element y ∈ R such that x²y = x. It results   x (xy - 1) = 0, with xy - 1 ≠0, since x is a non unit element in R. Then xy - 1 ∈ A and A ≠ {0}. We have that A ≠ R since x ∉ A (we have x² ≠ 0). From here, we obtain that A = I, a contradiction, therefore I² = {0}.□

Proposition 3.8. Each nonzero element in a finite commutative unitary ring R is a unit or a zero divisor.

Proof. Let R = {x₁, x₂, . . . , x _n } be a ring with n elements and a ∈ R a non unit element. Therefore, the following elements ax₁, ax₂, . . . , ax _n are all different from 1. It results that we can have maximum n - 1 distict values, then there are equal at least two of them. We otain that ax _i  = ax _j , for i ≠ j, therefore a (x _i  - x _j ) = 0. Since x _i  ≠ x _j , it results that a is a zero divisor. □

Remark 3.9. (see [13])

1. In a commutative ring R, the set of non-unit elements is an ideal if and only if the ring R is local.

2. If R is a commutative Noetherian ring and I is an ideal in R consists of zero-divisors, then its annihilator is a non zero ideal.

Proposition 3.10. Let R be an integral domain with a finite number of ideals. Then R is a field.

Proof. Let a ∈ R be a nonzero element. We have the ideals <a > , < a² > , . . . , < a ^k  > , . . .Since we have a finite number of ideals, it results that we can find two equal ideals in this chain: <a ^{k ₁} > = < a ^{k ₂}  >. Therefore, there is an element r ∈ R such that a ^{k ₁} r = a ^{k ₂} . Assuming that k₁ ≥ k₂, it results that a ^{k ₂}  (a^k₁-h₂r - 1) = 0. Since a ¬ =0, we have that a^k₁-h₂r - 1 =0, therefore a is an invertible element.

□

Proposition 3.11. Let R be a commutative and unitary ring with a finite number of ideals. Therefore, an ideal I is maximal if and only if it is a prime ideal.

Proof. Let I be a maximal ideal. Therefore, R/I  is a field, that means an integral domain, therefore I is a prime ideal. Conversely, if I is a prime ideal, therefore R/I is an integral domain with a finite number of ideals. From the above proposition, R/I is a field, then I is a maximal ideal. □

Proposition 3.12. Let R be a commutative and unitary ring with a finite number of ideals. Let n _m  (R) be the number of maximal ideals in R, n _p  (R) be the number of prime ideals in R and n _I  (R) be the number of all ideals in R. Therefore, n _m  (R) = n _p  (R) = α and n I ( R ) = ∏ α j = 1 β j , β j positive integers, β _j  ≥ 2.

Proof. Relation n _m  (R) = n _p  (R) = α arises from the above proposition. It is clear that R is an Artinian ring, therefore it is a product of local nonzero rings, R = R₁ × . . . × R _α. Therefore, each ideal J in R is of the form J = J₁ × . . . × J _α, with J _i ideal in R _i . Since each ring R _i has at least two ideals, we obtain that n I ( R ) = ∏ α j = 1 β j , where β _j  ≥ 2 is the number of ideals in R _j . □

Remark 3.13. 1) First, we must remark that n _I  (R _j ) can be any positive integer β _j  ≥ 2. For example, the ring K [X]/(X ^{β _j} ) has β _j ideals, β _j  ≥ 2.

2) If n _I  (R) is finite, from Proposition 2.11, we have only the following two possibilities:

-R is an integral domain, therefore it is a field and, in this case, we have n _m  (R) = n _p  (R) = 1 and n _I  (R) = 2 or

- R is not an integral domain and n _m  (R) = n _p  (R) ≥ 1 and n _I  (R) > 2.

3) From the above proposition it is clear that there are not commutative unitary rings such that (n _m  (R) , n _p  (R) , n _I  (R)) = (3, 3, 5) or (n _m  (R) , n _p  (R) , n _I  (R)) = (2, 2, 5), since 5 is a prime number. To find such an example, we must search in non-commutative rings. Therefore we have examples only in the case (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 5). The same situation for n _I  (R) = 7. We have only the case (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 7) But, we can find examples of commutative unitary rings such that (n _m  (R) , n _p  (R) , n _I  (R)) = (2, 2, 4) or commutative unitary rings such that (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 4). For example, for the first case, we have the ring ℤ 2 × ℤ 2 or ℤ 6 and for the second, we have the ring ℤ 8 .

4 Remarks regarding BL-rings

Let R be a commutative unitary ring and Id (R) be the set of all ideals of the ring R. We know that (Id (R) , ∩ , + , ⊗ → , 0 = {0} , 1 = R) is a residuated lattice in which the order relation is ⊆ and I → J = (J : I) , for every I, J ∈ Id (R). A commutative ring is a BL-ring if and only if Id (R) is a BL-algebra (see [9], Corollary 2.3.). A Noetherian multiplicative ring is an example of BL-ring (see [9], Example 2.4, 2.).

If we consider R₁ and R₂ two BL-rings and their lattices of ideals Id (R₁) and Id (R₂), therefore the lattice Id (R₁ × R₂) = Id (R₁) × Id (R₂) is a BL-algebra, then R₁ × R₂ is a BL-ring. Indeed, for Z, W ∈ Id (R₁ × R₂) , Z = (I₁, J₁) and W = (I₂, J₂), we define Z → W = ( I 1 → I 2 , J 1 → J 2 ) , and Z ⊗ W = ( I 1 ⊗ J 1 , I 2 ⊗ J 2 ) . In this way, we obtain a BL-algebra structure on Id (R₁ × R₂).

Theorem 4.1. There are no commutative unitary rings R with three ideals having (Id (R) , ∩ , + , ⊗ → , 0 = {0} , 1 = R) as a BL-algebra which is not an MV-algebra.

Proof. From Proposition 2.7, it results that if R is a commutative unitary ring which has exactly three ideals {0} , I, R, therefore we have I² = {0}. From here, we have I → 0 = (0 : I) = I and the following implication and multiplication tables are obtained

→ 0 I R 0 R R R I I R R R 0 I R ⊗ 0 I R 0 0 0 0 I 0 0 I R 0 I R(4.1) which give us an MV-algebra structure. An example of such a ring is ( ℤ 4 , + , · ) , with I = ( 2 ˆ ) , 0 = ( 0 ˆ ) .

□

Remark 4.2. From the above, a commutative unitary ring R with three ideals always has on the algebra Id (R) an MV-algebra structure. Therefore, to obtain rings with three ideals such that Id (R) is a BL-algebra that is not an MV-algebra, we must search among non-unitary rings or among non-commutative rings or both, or to find such an algebra as a subset or as a subalgebra of an infinite BL-algebra. The only BL-algebra of order 3 which is not an MV-algebra has the following implication and multiplication tables: → 0 I R 0 R R R I 0 R R R 0 I R ⊗ 0 I R 0 0 0 0 I 0 I I R 0 I R(4.2) From the above, it is clear there is not a commutative unitary ring R with three ideals such that the algebra Id (R) is a BL-algebra with → and ⊗ given by (3.2).

Remark 4.3. 1) We recall that if K is a field and K [X] is the polynomial ring, then for the polynomial f ∈ K [X], the quotient ring A = K [X]/(f) is a BL-ring, see [14].

2) Let n = p₁p₂, with p₁ and p₂ two prime distinct integers. We have the following isomorphism R = ℤ p 1 p 2 [ X ] ( p 1 X 2 + p 2 ) ≃ ℤ [ X ] ( n , p 1 X 2 + p 2 ) ≃ ℤ [ X ] ( p 2 , X 2 ) ≃ ℤ p 2 [ X ] ( X 2 ) . This ring is a BL-ring.

Example 4.4. In the following, we present some examples of BL-rings, more precisely, examples of Noetherian multiplicative rings which are BL-rings.

1) We consider the ring R = ℤ 6 [ X ] ( X 2 ) . From the above, we have ℤ 6 [ X ] ( X 2 ) ≃ ℤ 2 [ X ] ( X 2 ) × ℤ 3 [ X ] ( X 2 ) . The ring ℤ 2 [ X ] ( X 2 ) has three ideals (0) , (X) and ℤ 2 [ X ] ( X 2 ) . The ring ℤ 3 [ X ] ( X 2 ) has also three ideals (0) , (X) and ℤ 3 [ X ] ( X 2 ) Therefore, the ring ℤ 6 [ X ] ( X 2 ) is a BL-ring with 9 ideals.

2) From the above results, we have the following rings isomorphisms R 1 = ℤ 6 [ X ] ( 2 X 2 + 3 ) ≃ ℤ [ X ] ( 6 , 2 X 2 + 3 ) ≃ ℤ [ X ] ( 3 , X 2 ) ≃ ℤ 3 [ X ] ( X 2 ) and R 2 = ℤ 6 [ X ] ( 3 X 2 + 2 ) ≃ ℤ [ X ] ( 6 , 3 X 2 + 2 ) ≃ ℤ [ X ] ( 2 , X 2 ) ≃ ℤ 2 [ X ] ( X 2 ) .

Indeed, since X² = (2X² + 3) (3X² + 2), we have that (3,  X²) ⊆ (6,  2X² + 3). Conversely it is also true, since 2X² + 3 =2 · X² + 3 and 6 = 3 +3. Therefore (6,  2X² + 3) = (3,  X²). Similar, we have (6,  3X² + 2) = (2,  X²). In the rings R₁ and R₂, we have X² = 0, since X² = (2X² + 3) (3X² + 2). Then, in R₁ we have 3 = 0 and in R₂   we have 2 = 0 .

3) From the above results, we obtain the following isomorphism of rings: ℤ 6 [ X ] ( 2 X 2 + 4 ) ≃ ℤ [ X ] ( 6 , 2 X 2 + 4 ) ≃ ℤ [ X ] ( 2 ) ( 3 , X 2 + 2 ) ≃ ≃ ℤ [ X ] ( 2 ) × ℤ [ X ] ( 3 , X 2 + 2 ) ≃ ℤ 2 [ X ] × ℤ 3 [ X ] ( X 2 + 2 ) . Therefore, the ring ℤ 6 [ X ] ( 2 X 2 + 4 ) is an infinite BL-ring.

In the same way, we have the following isomorphisms of rings:

ℤ 30 [ X ] ( 6 X 2 + 24 ) ≃ ℤ [ X ] ( 30 , 6 X 2 + 24 ) ≃ ℤ [ X ] ( 6 ) ( 5 , X 2 + 4 ) ≃ ≃ ℤ [ X ] ( 6 ) × ℤ [ X ] ( 5 , X 2 + 4 ) ≃ ℤ 6 [ X ] × ℤ 5 [ X ] ( X 2 + 4 ) , or ℤ 30 [ X ] ( 6 X 2 ) ≃ ℤ [ X ] ( 30 , 6 X 2 ) ≃ ℤ [ X ] ( 6 ) ( 5 , X 2 ) ≃ ≃ ℤ [ X ] ( 6 ) × ℤ [ X ] ( 5 , X 2 ) ≃ ℤ 6 [ X ] × ℤ 5 [ X ] ( X 2 ) . Therefore, the rings ℤ 6 [ X ] × ℤ 5 [ X ] ( X 2 + 4 ) and ℤ 6 [ X ] × ℤ 5 [ X ] ( X 2 ) are infinite BL-rings.

4) From the above results, we have the following isomorphism of rings: ℤ 6 [ X ] ( X 2 - X ) ≃ ℤ 6 [ X ] X ( X - 1 ) ≃ ℤ 6 [ X ] ( X ) × ℤ 6 [ X ] ( X - 1 ) ≃ ≃ ℤ 6 × ℤ 6 .

Example 4.5. Let R = ℤ 6 [ X ] × ℤ 5 [ X ] ( X 2 ) , be a BL-ring from the above examples. Therefore Id (R) is an infinite BL-algebra. In R, we consider the following ideals {0} , I = (3) × R and R, where ( 3 ) = { f ∈ ℤ 6 [ X ] , f = a n X n + . . . + a 1 X + a 0 , with a₀ = 0 or a₀ = 3}. We have that I² = I and Ann (I) = {0}. Therefore, B = {{0} , I, R} is a finite BL-subalgebra of the algebra Id (R), having the implication and multiplication tables given by the relation (3.2 .).

Theorem 4.6. Let R be a commutative unitary ring which has exactly four ideals {0} , I, J, R. Therefore, the lattice Id (R) can be of the form A or B from Figure 1.

OPEN IN VIEWER

Fig. 1

Latices with four elements.

Proof.

Case i. I and J are maximal ideals.

In this situation, we have {0} ⊂ I ⊂ R and {0} ⊂ J ⊂ R, with I + J = R. Therefore, Id (R) is a lattice as in Figure 1, A. We have that I and J are coprime. It is clear that I ⊗ J = I ∩ J = {0} and I² = I, J² = J. From here, it results that Ann (I) = J and Ann (J) = I. Also, J → I = (I : J) = I and I → J = (J : I) = J. Therefore, we obtain an MV-algebra structure with the following operations:

→ 0 I J R 0 R R R R I J R J R J I I R R R 0 I J R ⊗ 0 I J R 0 0 0 0 0 I 0 I 0 I J 0 0 J J R 0 I J R(4.3)

Case ii. The lattice of ideals is a chain, Id (R) = {{0} ⊂ I ⊂ J ⊂ R}, as in Figure 1, B.

Since {0} ⊂ I ⊂ J ⊂ R, the ring is local. Also, we remark that I is the only minimal ideal. Let x ∈ I, x ≠ 0. Therefore, 0 ¬ = < x > ⊆ I. Since I is minimal, we have I = < x > , then I is finitely generated. We have the following subcases.

1) I² = {0} , J² = I, I ⊗ J = {0}. We have Ann (I) = (0 : I) = J and Ann (J) = (0 : J) = I. Also, J → I = (I : J) = J. Therefore, we obtain an MV-algebra structure with the following operations:

→ 0 I J R 0 R R R R I J R R R J I J R R R 0 I J R ⊗ 0 I J R 0 0 0 0 0 I 0 0 0 I J 0 0 I J R 0 I J R(4.4)

2) I² = {0} , J² = I, I ⊗ J = I.  In this case we do not obtaine a residuated lattice, since ⊗ is not associative. For example, I ⊗ J² = I² = {0} and (I ⊗ J) ⊗ J = I ⊗ J = I.

3) I² = {0} , J² = J, I ⊗ J = I. We have Ann (I) = (0 : I) = I and Ann (J) = (0 : J) = {0}. Also, J → I = (I : J) = I. Therefore, for Id (R)  we obtain a BL-algebra structure (which is not an MV-algebra) with the following implication and multiplication tables

→ 0 I J R 0 R R R R I I R R R J 0 I R R R 0 I J R ⊗ 0 I J R 0 0 0 0 0 I 0 0 I I J 0 I J J R 0 I J R(4.5)

4) I² = {0} , J² = J, I ⊗ J = {0}. We have Ann (I) = (0 : I) = J and Ann (J) = (0 : J) = I. Also, J → I = (I : J) = I. Therefore, we have J ⊗ (J → I) = J ⊗ I = {0} and J ∧ I = I, false. Condition (div) is not satisfied. It results that Id (R) is not a BL-algebra.

5) I² = {0} , J² = {0} , I ⊗ J = {0}. We have Ann (I) = (0 : I) = J and Ann (J) = (0 : J) = J. Also, J → I = (I : J) = J. Therefore, we have J ⊗ (J → I) = J ⊗ J = {0} and J ∧ I = I, false. Condition (div) is not satisfied. It results that Id (R) is not a BL-algebra.

6) I² = {0} , J² = {0} , I ⊗ J = I, it is not possible, since I = I ⊗ J ⊂ J² = {0}.

7) I² = I, J² = I, I ⊗ J = I. We have Ann (I) = (0 : I) = 0 and Ann (J) = (0 : J) = 0. Also, J → I = (I : J) = J. Therefore, for Id (R)  we obtain a BL-algebra structure(which is not an MV-algebra) with the following implication and multiplication tables:

→ 0 I J R 0 R R R R I 0 R R R J 0 J R R R 0 I J R ⊗ 0 I J R 0 0 0 0 0 I 0 I I I J 0 I I J R 0 I J R(4.6)

8) I² = I, J² = I, I ⊗ J = {0}, it is not possible, since I = I² ⊂ I ⊗ J = {0}.

9) I² = I, J² = J, I ⊗ J = I. We have Ann (I) = (0 : I) = 0 and Ann (J) = (0 : J) = 0. Also, J → I = (I : J) = I. Therefore, for Id (R)  we obtain a BL-algebra structure (which is not an MV-algebra) with the following implication and multiplication tables:

→ 0 I J R 0 R R R R I 0 R R R J 0 I R R R 0 I J R ⊗ 0 I J R 0 0 0 0 0 I 0 I I I J 0 I J J R 0 I J R(4.7)

10) I² = I, J² = J, I ⊗ J = {0}, it is not possible, since I = I² ⊂ I ⊗ J = {0}. 11) I² = I, J² = {0} , I ⊗ J = I, it is not possible, since I = I ⊗ J ⊂ J² = {0}. 12) I² = I, J² = {0} , I ⊗ J = {0}, it is not possible, since from I ⊂ J we obtain I² ⊂ J² = {0}, so I = {0} , a contradiction.

Remark 4.7. Counting the BL-algebras of order 4 from the above theorem, we obtain 5 BL-algebras of order 4, two MV and three BL-chain. In this way, we recover the results obtained in [15], Table 2. From the above remark, we see that all BL-algebras of order four can be obtained as a lattice of ideals of a commutative unitary ring R. For MV-algebra given by the relation (3.3), there is an example of such type of ring, namely R = ℤ 2 × ℤ 2 , and for MV-algebra given by the relation (3.4) an example can be the ring ℤ 8 (see [14], Table 2). It is interesting to search and find, if there are, examples of commutative unitary rings R with Id (R) a BL-algebra given by relations (3.5) , (3.6) , (3.7).

Remark 4.8. a) Other examples of BL-rings are presented in the following. From ([9], Example 2.4 and Remark 2.5) we know that a Noetherian multiplication ring is a BL-ring. The ring ℤ is a Noetherian multiplication ring (it is principal). Therefore, from the above properties ( ℤ n , + , · ) is a Noetherian multiplication ring.

b) We consider the ring ( ℤ n , + , · ) with n = p₁p₂ . . . p _r , p₁, p₂, . . . , p _r being distinct prime numbers, r ≥ 2. The ring R = ℤ n [ X ] / ( f ) , with f a polynomial mod n of degree q ≥ 2, is a Noetherian multiplication ring, therefore a BL-ring. From here, we have that the algebra of ideals, Id (R), are BL-algebra which is not a BL-chain ( ℤ n is a direct product of the fields ℤ p i ).

c) If n = p ^r , r ≥ 2, there is an MV-algebra isomorphic to ℤ n ([14]).

d) From ([10], Corollary 1.3), we know that an Artinian ring is a multiplication ring if and only if it is a finite product of Artinian local principal ideal rings. Moreover, if r = 1, the ring R = ℤ p [ X ] / ( f ) is Artinian (is finite), local and principal ring, therefore it is a BL-ring.

e) If we take the ring ℤ p r = ℤ p × . . . × ℤ p ︸ r - time , we have that the ring R = ℤ p r [ X ] / ( f ) is a Noetherian multiplication ring, therefore a BL-ring.

Theorem 4.9. Let R be a commutative unitary ring which has exactly five ideals. Therefore, the lattice Id (R) can be of the form A-E from Figure 2 and Figure 3.

Proof. Indeed, since 5 is a prime number, with the above notations, we have only commutative unitay rings R of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 5) with Id (R) a BL-algebra, that means Id (R) is a chain (Figure E) or has Hasse diagram as in Figure C. Also, we remark that a BL-algebra cannot have the form as in Figure C since, in this case (a → b) ∨ (b → a) = c ≠ 1, so (prel) condition is not satisfied. We conclude that Id (R) is a chain.

In [1], Corollary 28, the authors proved that each finite BL-algebras are isomorphic to a direct product of BL-comets. A BL-comet of order 5 has Hasse diagram as in Figures 3, D or E. If we consider a BL-comet of order 5 as in Figure 3, D, generated by a commutative ring R, then Id (R) has two maximal ideals, impossible. We conclude that, BL-comets Id (R) generated by commutative rings R with five ideals are chains. As in Remark 3.6, we can obtain all 8 BL-chain of order 5, one MV and seven BL-chain as Id (R) of a ring with 5 ideals with (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 5). In this way, we recover the results obtained in [15], Table 2.

OPEN IN VIEWER

Fig. 2

Latices with five elements.

OPEN IN VIEWER

Fig. 3

Latices with five elements.

Remark 4.10. a) With the above notations, we remark that we do not have examples of commutative unitary rings R of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (2, 2, 6) with Id (R) a BL-algebra which is not an MV-algebra. Indeed, if such a ring exists, it is an Artinian ring, therefore it is a product of Artinian local nonzero rings, R = R₁ × R₂. One of these rings, R₁, has two ideals, and the other one, R₂, has three ideals. Therefore Id (R₁) and Id (R₂) must be BL-algebras and at least one should not be an MV-algebra.. But, the only BL-algebra with 2 elements, Id (Z₂) , is an MV-algebra and from Proposition 3.1, we have that there are not commutative unitary rings with three ideals such that Id (R₂) is a BL-algebra which are not an MV-algebra. In this case we have only finite commutative unitay rings R of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 6) with Id (R) a BL-algebra which is not an MV-algebra.

b) We can have only commutative unitay R rings of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 7) with Id (R) a BL-algebra, since 7 is a prime number, that means Id (R) is a chain.

b) If n _I  (R) = 8, we can have commutative unitary rings of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (2, 2, 8)  or (n _m  (R) , n _p  (R) , n _I  (R)) = (3, 3, 8), with Id (R) a BL-algebra.

d) For n _I  (R) = 9, we do not have commutative unitary rings of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (2, 2, 9), with Id (R) a BL-algebra which is not an MV-algebra, since there are not commutative unitary rings R with three ideals such that Id (R) is a BL-algebra which is not an MV-algebra. But we can have rings R of the form (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 9), that means Id (R) is a chain. For MV-algebras of order 9, there are examples of rings whit which this MV-algebras is isomorphic to, namely ℤ 3 × ℤ 3 or ℤ 9 (see [14], Table 2).

e) For n _I  (R) = 10, we can have (n _m  (R) , n _p  (R) , n _I  (R)) = (2, 2, 10) or (n _m  (R) , n _p  (R) , n _I  (R)) = (1, 1, 10) such that Id (R) is a BL-algebra which is not an MV-algebra.

Example 4.11. Let R be a BL ring with Id (R) infinite and I be an idempotent ideal in R, I² = I. From Example 3.5, we know that such a ring exists and {0, I, R} forms a BL-subalgebra of order 3 with the multiplication tables given by the relation (3.2). We consider the ring R × R, therefore B = { ( 0 , 0 ) , ( 0 , I ) , ( 0 , R ) , ( I , 0 ) , ( I , I ) , ( I , R ) , ( R , 0 ) , ( R , I ) , ( R , R ) } = { O , A , B , C , D , E , F , G , Z }  is a BL-subalgebra of order 9 (see Figure 4). We have the following implication and multiplication tables:

→ O A B C D E F G Z O Z Z Z Z Z Z Z Z Z A F Z Z F Z Z F Z Z B F G Z F G Z F G Z C B B B Z Z Z Z Z Z D O B B F Z Z F Z Z E O A B F G Z F G Z F B B B E E E Z Z Z G O B B C E E F Z Z Z O A B C D E F G Z ⊗ O A B C D E F G Z O O O O O O O O O O A O A A O A A O A A B O A B O A B O A B C O O O C C C C C C D O A A C D D C D D E O A B C D E C D E F O O O C C C F F F G O A A C D D F G G Z O A B C D E F G Z(4.8)

OPEN IN VIEWER

Fig. 4

Latices with nine elements.

From here, we remark that {C, D, E, F, G, Z} is a BL-algebra of order 6 (see bold parts from relation (3.8)), extracted from the algebra B , {O, D, E, G, Z} is a BL-subalgebra of order 5 (it is a BL-comet, from the below tables) and {F, G, Z} is a BL-algebra of order 3 (as in relation (3.7)), extracted from the algebra B (see Figure 5).

→ O D E G Z O Z Z Z Z Z D O Z Z Z Z E O G Z G Z G O E E Z Z Z O D E G Z ⊗ O D E G Z O O O O O O D O D D D D E O D E D E G O D D G G Z O D E G Z

OPEN IN VIEWER

Fig. 5

Latices with six elements.

In this paper, we presented some conditions for rings with a finite number of ideals to have their lattice of ideals equipped with a structure of BL-algebra.

As a further research, we intend to find circumstances for a given ring R  to have on its lattice of ideals Id (R) a BL-algebra structure which is not an MV-algebra. As we have seen in the above, these conditions depend on the given number of ideals, that means depend on the 3-uple (n _m  (R) , n _p  (R) , n _I  (R)). Since it is clear that there are some values which indicate that we must search amoung non-unitary rings or amoung non-commutative rings or both, this study can helps us to give a classification of this type of rings.

Acknowledgments

The authors thank referees for thier patience and for their advices which helped us to improve this paper.

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