PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 87–93 DOI: 10.2298/PIM1614087S

DISTRIBUTIVE LATTICES OF JACOBSON RINGS

Yong Shao, Siniša Crvenković, and Melanija Mitrović

Abstract. We characterize the distributive lattices of Jacobson rings and prove that if a semiring is a distributive lattice of Jacobson rings, then, up to isomorphism, it is equal to the subdirect product of a distributive lattice and a Jacobson ring. Also, we give a general method to construct distributive lattices of Jacobson rings.

1. Introduction and preliminaries

A semigroup S is called periodic if each element of S has a finite order, where the order of a ∈ S is the order of the cyclic subsemigroup of S generated by a. Periodic semigroups have been studied by many algebraists. Suppose that S is a periodic semigroup. For any a ∈ S we all know that there exist the smallest positive integer m and the smallest positive integer r such that a^{m} = a^{m+r} . The positive integer m is referred to as the index and the positive integer r as the period of a. In particular, if the index of each a ∈ S is equal to 1, then we call S a strongly periodic semigroup. Idempotent semigroups and Burnside semigroups satisfying x^{n} ≈ x are special cases of strongly periodic semigroups.

A ring (R, +, ·) is a Jacobson ring if, for any a ∈ R, there exists n ∈ ℕ, n ⩾ 2 such that a = a^{n}. That is to say, its multiplicative reduct is a strongly periodic semigroup. It is obvious that Boolean rings are Jacobson rings. Following [6, Theorem 11], we have

Theorem 1.1. Let R be a Jacobson ring. Then every element of R has finite additive order and R is commutative.

We denote by JR the class of all Jacobson rings.

2010 Mathematics Subject Classification: Primary 16Y60, 08B05; Secondary 20M07.

Key words and phrases: semiring; distributive lattice; Jacobson ring; Mal’cev product; congruence.

The first author is supported by the Natural Science Foundation of Shannxi Province, Grant 2015JQ1020. The second author is Supported by the Ministry of Education, Science and Technological Development of Serbia, Grant 174018. The third author is Supported by the Ministry of Education, Science and Technological Development of Serbia, Grant 174026.

Communicated by Žarko Mijajlović.

Semirings are the natural generalization of rings and distributive lattices. All semirings (S, +, ·) occurring in the literature satisfy at least the following axioms: (S, +), the additive reduct, and (S, ·), the multiplicative reduct of a semiring S are semigroups, and the multiplication distributes over addition from both sides, i.e.,

(SR1) x + (y + z) ≈ (x + y) + z;

(SR2) x(yz) ≈ (xy)z;

(SR3) x(y + z) ≈ xy + xz, (x + y)z ≈ xz + yz.

It is, as well, often assumed that (S, +) is commutative, i.e., (SR4) x + y ≈ y + x.

We consider the semiring classes considered that satisfy this identity too. By an idempotent semiring, we mean a semiring S which satisfies the addi-

tional identities xx ≈ x + x ≈ x. An idempotent semiring (S, +, ·) is called a bisemilattice if both the multiplicative reduct (S, ·) and the additive reduct (S, +) are semilattices. Of course, a distributive lattice is a bisemilattice which satisfies the absorption law x + xy ≈ x. The class of all distributive lattices is denoted by D. The Mal’cev product of two classes V and W of semirings, denoted by V ∘ W, we mean that the class of all semirings S on which there exists a congruence ρ such that S/ρ ∈ W and the ρ-classes which are subsemirings of S belong to V. Thus, in this way, some classes of semirings can be constructed by considering the Mal’cev products of some given semirings.

For a semiring (S, +, ·) we denote Green’s H relation on the additive reduct (S, +) by H^{+} . Let (S, +, ·) be a semiring whose additive reduct (S, +) is a completely regular semigroup. By Theorem II.1.4 and Corollary II.1.5 in [8], (S, +) is a commutative Clifford semigroup and H^{+} is the least semilattice congruence of the additive reduct (S, +) of S, moreover, every H^{+} -class is a maximal subgroup of (S, +). For any a ∈ S we denote by H_{a}^{+} the H^{+} -class containing a and 0_{a} the identity of H_{a}^{+} , respectively. It can be easily seen that aH^{+} b if and only if 0_{a} = 0_{b} for any a, b ∈ S.

Let (S, +, ·) be a semiring whose additive reduct is a Clifford semigroup. We can define the natural partial order on (S, +) by

a ⩽_{+} b ⇔ (∃e ∈ E_{+} (S)) a = b + e

for a, b ∈ S, where E_{+} (S) is the set of idempotents of the additive reduct (S, +) of S.

The Mal’cev product of the class of Jacobson rings and the class of distributive lattices is denoted by JR ∘ D. A semiring S is called a distributive lattice of Jacobson rings if it is in JR ∘ D. In the following we shall study the semirings which are distributive lattices of Jacobson rings.

Some authors have studied the distributive lattices of rings(see [1, 2, 7]). In particular, [1] and [2] characterized the subdirect product of rings and distributive lattice, respectively. If a semiring (S, +, ·) is isomorphic to a subdirect product of a ring and a distributive lattice, then the additive reduct (S, +) of S is a sturdy semilattice of abelian groups, which means that (S, +) is E-unitary. The following example shows that, in general, distributive lattices of rings are not the subdirect product of a ring and a distributive lattice.

Example 1.1. Consider a five element semiring A_{5} with operations given by

\begin{matrix}+ & a & b & c & d & e \\ a & a & b & a & b & e \\ b & b & a & b & a & e \\ c & a & b & c & d & e \\ d & b & a & d & c & e \\ c & e & e & e & e & e \end{matrix}  \begin{matrix}. & a & b & c & d & e \\ a & a & a & c & c & a \\ b & a & a & c & c & a \\ c & c & c & c & c & c \\ d & c & c & c & c & c \\ e & a & a & c & c & e \end{matrix}

It is easy to see that H^{+} is a distributive lattice congruence and H_{a}^{+} = {a, b}, H_{c}^{+} = {c, d} and H_{e}^{+} = {e} are subrings of A_{5}. Since e ⩽_{+} a, e ⩽_{+} b, this means that (A_{5}, +) is not E-unitary. In [2] and [4], it was proved that a distributive lattice of Boolean rings is isomorphic to subdirect product of a Boolean ring and a distributive lattice. In this paper we prove that if a semiring is distributive lattice of Jacobson rings, then, up to isomorphism, it is equal to the subdirect product of a distributive lattice and a Jacobson ring, which generalize and enrich some results from [2, 4, 10]. Also, we shall give a general method to construct distributive lattices of Jacobson rings.

2. Main results

Lemma 2.1. A semiring S is a distributive lattice of Jacobson rings, i.e., S ∈ JR ∘D, if and only if H^{+} is the least distributive lattice congruence on S and every H^{+} -class is a Jacobson ring.

Proof. Let a semiring S be a distributive lattice of Jacobson rings. Then there exists a semiring congruence ρ on S such that S/ρ is a distributive lattice and every ρ-class is a Jacobson ring. This also implies that ρ is a semilattice congruence on (S, +). Since the additive reduct (S, +) of S is a Clifford semigroup, H^{+} is the least semilattice congruence on (S, +). This leads to H^{+} ⊆ ρ. On the other hand, since ρ_{u} (the ρ-class containing u) is a Jacobson ring for any u ∈ S, the additive reduct of ρ_{u} is an abelian subgroup of (S, +). Thus ρ_{u} ⊆ H_{u}^{+} , furthermore, ρ ⊆ H^{+} . We have now shown that ρ = H^{+} . That is to say that H^{+} is a distributive lattice congruence of semiring S and every H^{+} -class is a Jacobson ring.

If μ is a distributive lattice congruence on S, then μ is a semilattice congruence on the additive reduct (S, +). Since H^{+} is the least semilattice congruence on (S, +), H^{+} ⊆ μ, which implies that H^{+} is the least distributive lattice congruence on S.

Conversely, it is clear from definition that the semiring S is a distributive lattice of Jacobson rings since H^{+} is the least distributive lattice congruence on the semiring S and every H^{+} -class is a Jacobson ring. □

As a consequence of Lemma 2.1 we have the following result.

Corollary 2.1. Let S be a semiring in JR ∘ D. Then

(i) for any a, b ∈ S, 0_{a} + 0_{b} = 0_{a+b} , a0_{b} = 0_{b}a = 0_{a}0_{b} = 0_{ab}, a + a0_{b} = a; (ii) E_{+} (S) = {0_{a} | a ∈ S} is a distributive lattice.

Let S be a semiring in JR ∘ D. Define a binary relation σ^{+} on S by

(∀a, b ∈ S) a σ^{+} b ⇔ (∃e ∈ E_{+} (S)) a + e = b + e.

It follows from Proposition 5.3.2 in [5] that σ^{+} is the least group congruence on the additive reduct (S, +) of S. Thus we have

Lemma 2.2. Suppose that S is a semiring in JR ∘ D. Then σ^{+} is the least Jacobson ring congruence on S.

Proof. Assume that a, b ∈ S and a σ^{+} b. Then there exists e ∈ E_{+} (S) such that a + e = b + e. For any c ∈ S we have that ca + ce = cb + ce. By Corollary 2.1 it follows that ce ∈ E_{+} (S) and so ca σ^{+} cb. Dually, we can get ac σ^{+} bc. Thus, σ^{+} is a semiring congruence on S. Since (S/σ^{+} , +) is an abelian group, (S/σ^{+} , +, ·) is a ring. For any a ∈ S we denote by σ_{a}^{+} the σ^{+} -class containing a. By Lemma 2.1 it yields that H_{a}^{+} is a Jacobson ring. Thus there exists a positive integer k such that a^{k} = a. Therefore, (σ_{a}^{+} )^{k} = σ_{a^{k}}^{+} = σ_{a}^{+} . Hence, (S/σ^{+} , +, ·) is a Jacobson ring and so σ^{+} is a Jacobson ring congruence on S.

Suppose that θ is a Jacobson ring congruence on S. If a, b ∈ S and a σ^{+} b, then there exists f ∈ E_{+} (S) such that a + f = b + f. This yields θ_{a+f} = θ_{b+f} . Thus

θ_{a} = θ_{a} + θ_{f} = θ_{b} + θ_{f} = θ_{b}

since (S/θ, +) a group and and θ_{f} is the identity of (S/θ, +). This implies a θ b and so σ^{+} ⊆ θ. This shows that σ^{+} is the least Jacobson ring congruence on S. □

Now we are able to obtain the decomposition theorem of distributive lattice of Jacobson rings.

Theorem 2.1. Suppose that S is a semiring. Then S is a distributive lattice of Jacobson rings if and only if S is (isomorphic to) the subdirect product of a distributive lattice and a Jacobson ring.

Proof. Suppose that a ∈ S, e ∈ E_{+} (S) and a + e ∈ E_{+} (S). Thus there is f ∈ E_{+} (S) such that a+e = f. This yields a+e+f = e+f, and (left-)multiplying it by a, we have a^{2} + a(e + f) = a(e + f), which implies

(1)  a^{2} + a + a(e + f) = a + a(e + f).

Since the H^{+} -class containing a(e + f) is a Jacobson ring, by Theorem 1.1, there exists a positive integer k such that k ·(a(e+f)) = 0_{a(e+f)} . Adding (k −1)·a(e+f) to the both sides of (1), we get a^{2} + a + k · a(e + f) = a + k · a(e + f). This implies a^{2} + a + 0_{a(e+f)} = a + 0_{a(e+f)} . By Corollary 2.1(i), it follows that a^{2} + a = a, and, multiplying it by a, we have a^{3} + a^{2} = a^{2}, which implies a + a^{3} + a^{2} = a + a^{2}. Thus a^{3} + a = a. By induction, it can be easily shown that a^{m} + a = a for any positive integer m ⩾ 2. Since the H^{+} -class containing a is also a Jacobson ring, there exists a positive integer l ⩾ 2 such that a^{l} = a. Thus, a = a^{l} + a = a + a, i.e., a ∈ E_{+} (S). Therefore the additive reduct (S, +) is E-unitary. By Proposition 5.9.1 in [5] we have σ^{+} ∩ H^{+} = 1_{S}, which, by Lemma I.4.18 in [8], implies that S is the subdirect product of S/H^{+} and S/σ^{+} .

The converse is trivial. □

Let F_{1}, …, F_{k} be a fixed list of finite fields with different characteristics p_{1}, ..., p_{k} and respective sizes q_{1} = p_{1}^{n_{1}} , …, q_{k} = p_{k}^{n_{k}} , for some positive integers n_{1}, …, n_{k}. Let c = p_{1} ⋯ p_{k}, and let n be a positive integer such that n − 1 is the least common multiple of q_{1} − 1, …, q_{k} − 1. It was proved in [10] that the semiring variety V = HSP {B_{2}, F_{1}, …, F_{k}} generated by two-element distributive lattice B_{2} and finite fields F_{1}, …, F_{k} is determined by (SR1-4) and the following identities:

(DFSR1) (c + 1) · x ≈ x; (DFSR4) x + c · xy ≈ x;
(DFSR2) x^{n} ≈ x;  (DFSR5) xy ≈ yx;
(DFSR3) c · x^{2} ≈ c · x; (DFSR6) \frac{c}{p_{i}} · x^{q_{i}} ≈ \frac{c}{p_{i}} · x (1 ⩽ i ⩽ k).

Suppose that S is a semiring in V. From (SR4) and (DFSR1) we have that the additive reduct (S, +) is a commutative Clifford semigroup. It follows by Theorem 2.1 in [10] that S is isomorphic to the subdirect product of the distributive lattice S/H^{+} and Jacobson ring S/σ^{+} . Thus, by Theorem 2.1, S belongs to JR ∘D and so V ⊆ JR ∘ D. This shows that the above theorem generalizes Theorem 2.1 in [10].

In the rest of this section we give a method to construct distributive lattices of Jacobson rings. Assume that (D, +, ·) is a distributive lattice. Define a binary relation ⩽ on D by

(∀α, β ∈ D) α ⩽ β ⇔ α = α + β.

It is easy to check that ⩽ is a partial order on D. For any α, β ∈ D it is easy to see that α + β ⩽ α. Similarly, we have α + β ⩽ β. It is well known that α ⩽ αβ, β ⩽ αβ and α + β ⩽ αβ.

In order to discuss the structure of S, we have to recall the following concept from [9] and [11].

Let {(S_{α} , +, ·) | α ∈ D} be a family of disjoint semirings (S_{α} , +, ·) which are indexed by a distributive lattice D together with a family of monomorphisms φ_{α,β} : S_{α} → S_{β}(β ⩽ α) satisfying the following conditions: for any α, β, γ ∈ D,

(i) φ_{α,α} = 1_{S_{α}} ; (ii) If γ ⩽ β ⩽ α, then φ_{α,β} φ_{β,γ} = φ_{α,γ} ; (iii) If γ ⩽ α + β then

S_{α} φ_{α,γ} + S_{β}φ_{β,γ} ⊆ S_{α+β} φ_{α+β,γ} . S_{α} φ_{α,γ} · S_{β}φ_{β,γ} ⊆ S_{αβ} φ_{αβ,γ} .

On the set S = _{⋃ α∈D} S_{α} define addition and multiplication by

a + b = aφ_{α,α+β} + bφ_{β,α+β} ,
a · b= (aφ_{α,α+β} bφ_{β,α+β} )φ_{αβ,α+β}^{−1} ,

for any a ∈ S_{α} , b ∈ S_{β}. Then we can check that (S, +, ·) is a semiring, denoted by S = [D; S_{α} , φ_{α,β} ]. We call the constructed semiring S = [D; S_{α} , φ_{α,β} ] the sturdy distributive lattice of semirings S_{α}.

If all semirings S_{α} are in a class of semirings C, we call S = [D; S_{α} , φ_{α,β} ] the sturdy distributive lattice of C-semirings.

Theorem 2.2. Suppose that S is a semiring. Then S is a distributive lattice of Jacobson rings if and only if S is a sturdy distributive lattice of Jacobson rings.

Proof. Let a semiring S belong to JR ∘ D. By Lemma 2.1, we can assume that S is a distributive lattice D of Jacobson rings R_{α} ’s, where D ≅ S/H^{+} and each R_{α} is an H^{+} -class of S. For convenience, for any α ∈ D we denote by 0_{α} the unique idempotent of abelian group (R_{α} , +). Thus, E_{+} (S) = {0_{α} | α ∈ D}. From Lemma 2.2 we have that σ^{+} is the least Jacobson ring congruence on S, which means that (S/σ^{+} , +, ·) is a Jacobson ring. By Theorem 2.1 it follows that S is isomorphic to the subdirect product of the distributive lattice S/H^{+} and Jacobson ring S/σ^{+} . This implies that the additive reduct (S, +) of S is isomorphic to the subdirect product of the semilattice (S/H^{+} , +) and abelian group (S/σ^{+} , +). Thus, (S, +) is a sturdy semilattice (D, +) of abelian groups (R_{α} , +)(α ∈ D). Then, by Theorems IV.1.3, IV.1.6 and IV.1.7 in [8], we can express (S, +) = [(D, +); (R_{α} , +); φ_{α,β} ] as a sturdy semilattice of additive abelian groups R_{α} (α ∈ D), where (D, +)[(R_{α} , +)] denotes the additive semigroup of distributive lattice D [of Jacobson rings R_{α}] and φ_{α,β} is defined by

(∀a ∈ R_{α}) aφ_{α,β} = a + 0_{β}.
From (S, +) = [(D, +); (R_{α} , +); φ_{α,β} ] we have that φ_{α,β} (β ⩽ α) is a group

monomorphism from (R_{α} , +) to (R_{β} , +). In the following, we are going to show that φ_{α,β} (β ⩽ α) is a semiring homomorphism.

For a, b ∈ R_{α}, we have aφ_{α,β} = a + 0_{β} and bφ_{α,β} = b + 0_{β}. Then, by Corollary 2.1, we have

(ab)φ_{α,β} = ab + 0_{β} = a + a0_{β} + b0_{β} + 0_{β} = (a + 0_{β})(b + 0_{β}) = (aφ_{α,β} )(bφ_{α,β} ).

This shows that φ_{α,β} is a semigroup homomorphism from (R_{α} , ·) to (R_{β}, ·) and so φ_{α,β} is a semiring monomorphism.

For any α, β ∈ D, since R_{α} and R_{β} are H^{+} -classes and H^{+} is a distributive lattice congruence, R_{α} · R_{β} ⊆ R_{αβ} . Thus, for any a ∈ S_{α} , b ∈ S_{β}, we have

aφ_{α,α+β} = a + 0_{α+β} , bφ_{β,α+β} = b + 0_{α+β} , (ab)φ_{αβ,α+β} = ab + 0_{α+β} .

By Corollary 2.1 we have

ab + 0_{α+β} = a + a0_{α+β} + b0_{α+β} + 0_{α+β}
= (a + 0_{α+β} )(b + 0_{α+β} ) = (aφ_{α,α+β} )(bφ_{β,α+β} ).

Thus, (ab)φ_{αβ,α+β} = (aφ_{α,α+β} )(bφ_{β,α+β} ).

Let γ ∈ D and γ ⩽ α + β. Since φ_{α+β,γ} is a semiring homomorphism, we have

aφ_{α,γ} bφ_{β,γ} = aφ_{α,α+β} φ_{α+β,γ} bφ_{β,α+β} φ_{α+β,γ}
= (aφ_{α,α+β} bφ_{β,α+β} )φ_{α+β,γ} = (ab)φ_{αβ,α+β} φ_{α+β,γ} = (ab)φ_{αβ,γ} .

This shows R_{α} φ_{α,γ} · R_{β}φ_{β,γ} ⊆ R_{αβ} φ_{αβ,γ} . Hence,

ab = ((ab)φ_{αβ,α+β} )φ_{αβ,α+β}^{−1} = (aφ_{α,α+β} bφ_{β,α+β} )φ_{αβ,α+β}^{−1} .

Since a + b = aφ_{α,α+β} + bφ_{β,α+β} is evident, by the above definition, S is a sturdy distributive lattice D of Jacobson rings R_{α} ’s, where D ≅ S/H^{+} and each R_{α} is a H^{+} -class of semiring S.

Conversely, if the semiring S is a sturdy distributive lattice D of Jacobson rings R_{α} (α ∈ D), then S = [D; R_{α} , φ_{α,β} ]. Define a binary relation η on S by

(a, b ∈ S) a η b ⇔ (∃α ∈ D) a, b ∈ R_{α} .

It is a routine matter to verify that η is a distributive lattice congruence and that every η-class is a Jacobson ring. That is to say, S ∈ JR ∘ D. □

By Theorems 2.1 and 2.2 the following corollary is directly obtained.

Corollary 2.2. Let S be a semiring. Then the following statements are equivalent:

(i) S is a distributive lattice of Jacobson rings; (ii) S is the subdirect product of a distributive lattice and a Jacobson ring; (iii) S is a sturdy distributive lattice of Jacobson rings.

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School of Mathematics (Received 31 08 2015) Northwest University (Revised 21 01 2016) Xian, P.R. China

yongshaomath@126.com

Department of Mathematics and Informatics University of Novi Sad, Serbia

sima@eunet.rs

Faculty of Mechanical Engineering University of Niš, Serbia

meli@masfak.ni.ac.rs