Morita contexts , ideals , and congruences for semirings with local units

We consider Morita contexts for semirings that have certain local units but not necessarily an identity element. We show that the existence of a Morita context with unitary bisemimodules and surjective maps implies that the two semirings involved have isomorphic quantales of ideals and lattices of congruences.


INTRODUCTION
In the classical case [1;2, Chapter 6], Morita equivalence is an equivalence relation on the class of rings with identity, where two rings are considered equivalent if the categories of left (equivalently, right) modules over them are equivalent.This equivalence of categories turns out to be equivalent to the existence of a Morita context -a pair of bimodules over the two rings together with a pair of bimodule homomorphisms from their tensor products onto the original rings, satisfying certain conditions.While the definition of a Morita context may seem complex at first, it is often easier to prove statements about Morita equivalence using them rather than the categorical definition.
There have been several successful attempts to generalize Morita equivalence to settings other than rings with identity.Many results have be proven for rings with various kinds of local units [3,4].In this case, unitary modules are considered instead of arbitary modules in the definition of Morita equivalence as well as in Morita contexts.
Generalizing in another direction, rings and modules have been replaced with semigroups and acts over them.For monoids [5,6], Morita equivalence turns out to be very close to isomorphism and thus not very interesting.However, using local unit conditions like those for rings, as well as unitary acts instead of arbitary ones, gives a meaningful theory for semigroups where several results analogous to those of rings hold [7].
A semiring is an algebraic structure where the additive structure in the definition of a ring has been changed from an Abelian group to a monoid.The analogues for modules of rings are called semimodules.It is a natural question whether a Morita theory could be developed for semirings, and whether it is closer to the theory for rings or semigroups.For semirings with identity, Morita equivalence was first studied by Katsov and Nam [8] and further by Sardar, Gupta, and Saha [9][10][11].Morita equivalence for semirings with local units was first considered by Liu [12].
In this article, we approach Morita theory for semirings with local units from a different direction: that of Morita contexts.The relationship between Morita equivalence and the existence of a Morita context in this case has not been studied yet.We show, however, that the existence of a Morita context with conditions analogous to those used for rings and semigroups with local units implies that the two semirings have isomorphic lattices of ideals and congruences.These results are analogous to those obtained for semigroups with local units by Laan and Márki in [13] and for semirings with identity by Sardar and Gupta in [10].

DEFINITIONS Definition 1.
A semiring [14] is an algebra (S, +, •, 0) such that (S, +, 0) is a commutative monoid, multiplication is associative and distributes over addition from both sides, and 0 is a zero element with respect to multiplication.
Note that we do not require the existence of a multiplicative identity element.Golan [14] uses the term hemiring for the above definition and reserves semiring for semirings with identity.Definition 2. A left semimodule over a semiring S is an algebra S M = (M, +, 0, (s•)| s∈S ) such that (M, +, 0) is a commutative monoid and the following identities hold for all s, s ′ ∈ S, m, m ′ ∈ M: Right semimodules are defined analogously.Since a semiring is a semimodule over itself, Definition 4 also defines the product of two subsets of a semiring.This multiplication of subsets of a semiring is easily seen to be associative.

Definition 5. For a semiring S, a left (right) S-semimodule M is unitary if SM = M (MS = M). For semirings S and T , a bisemimodule S M T is unitary if S M and M T are unitary.
The following local unit conditions are chosen to cover an as large as possible class of semirings in the results to be proven.Both are implied by the notion of local units in [4, Definition 1].Definition 6.A semiring S has weak local units if for every s ∈ S there exist e, e ′ ∈ S with es = s = se ′ .Definition 7. A semiring S has common joint weak local units if for every s, s ′ ∈ S there exist e, e ′ ∈ S with s = ese ′ and s ′ = es ′ e ′ .Definition 8.An ideal of a semiring S is a set I ⊆ S that is a submonoid of (S, +) and for which SI ⊆ I and IS ⊆ I. Finitely generated ideals are defined as in ring theory.

Definition 9. A quantale is a complete lattice endowed with an associative multiplication that is distributive from both left and right with respect to joins of any cardinality. An isomorphism of quantales is a bijection from one quantale to another that preserves joins and meets of any cardinality and multiplication.
It is a well-known fact that the lattice Id(S) of ideals of a ring forms a quantale (see e.g.[15, p. 17]), where the multiplication of two ideals is given by Definition 4. It is easy to verify that the same fact holds for semirings.
Definition 10.For S-semimodules M S and S N, their tensor product M ⊗ N is defined as the factor semigroup of the free commutative additive semigroup F = F(M × N) generated by the set M × N, factorized by the congruence ρ generated by all ordered pairs of the form Note that the elements m ⊗ n form a system of generators for the semigroup M ⊗ N, i.e. every element of M ⊗ N is a finite sum of such elements.From the generating pairs of ρ we obtain the following basic identities: The semigroup M ⊗ N is actually a monoid, the zero element being 0 M ⊗ 0 N : The tensor product of semimodules was first introduced in [16] and further studied in [17].The following proposition can be proven as described in the paragraph preceding Theorem 3.1 of [17].

Proposition 1. Let R, S, and T be semirings and S M R and R N T bisemimodules. Then the monoid M R ⊗ R N can be turned in a unique way into a bisemimodule S M R ⊗ R N T , retaining its addition and zero element, such that for any m
Definition 11.A Morita context is a sextuple (S, T, S P T , T Q S , θ , ϕ ) where 1. S and T are semirings; 2. S P T and T Q S are bisemimodules as indicated by the subscripts; 3. θ : S (P ⊗ Q) S → S S S and ϕ : T (Q ⊗ P) T → T T T are bisemimodule homomorphisms; 4. for every p, p ′ ∈ P, θ (p ⊗ q)p ′ = pϕ (q ⊗ p ′ ); 5. for every q, q ′ ∈ Q, ϕ (q ⊗ p)q ′ = qθ (p ⊗ q ′ ).
We say that a Morita context (S, T, S P T , T Q S , θ , ϕ ) is unitary if S P T and T Q S are unitary bisemimodules.
Example.We give an example (inspired by the proof of [18,Theorem 9]) of a unitary Morita context with surjective mappings where the semirings are non-isomorphic.Let F be a free semiring with two generators x and y.Let ρ be the congruence on F generated by the pair (y, y 2 ), and let R := F/ρ.Then e := y/ρ is an idempotent.Let S be the subsemiring ReR of R; then S = SeS ̸ = eSe.Now one can verify that (S, eSe, S Se eSe , eSe eS S , θ , ϕ ), where θ (se ⊗ es ′ ) := ses ′ and ϕ (es ⊗ s ′ e) := ess ′ e, is a unitary Morita context with surjective mappings.

RESULTS
Our first result concerns ideals.The proof is analogous to that of Theorem 3 in [13] or Theorem 2.2 in [10].
Theorem 1.If two semirings S and T have weak local units and there exists a unitary Morita context (S, T, S P T , T Q S , θ , ϕ ) with θ , ϕ surjective, then there is a quantale isomorphism Id(S) → Id(T ) that takes finitely generated ideals to finitely generated ideals.
Proof.Let (S, T, S P T , T Q S , θ , ϕ ) be a unitary Morita context with θ , ϕ surjective.Define It is easily seen that the sets on the right side are indeed ideals.We show that Θ and Φ are mutually inverse bijections.Due to symmetry, it suffices to show that Θ(Φ(I)) = I for any I ∈ Id(S).First, To see this, observe that, according to Definitions 4 and 10, the subset of S on the left side consists of all finite sums of elements of the form θ (pϕ (qs ⊗ p ′ ) ⊗ q ′ ), where s ∈ I, p, p ′ ∈ P, q, q ′ ∈ Q.This transforms into θ (pϕ (qs and elements of this form generate the set on the right side.Now θ (P ⊗ Q)Iθ (P ⊗ Q) = SIS ⊆ I. Using weak local units, we can see that I ⊆ SIS, concluding the proof that Θ and Φ are mutually inverse.It is easy to see that for J ′ ⊆ J, Θ(J ′ ) ⊆ Θ(J) and the same for Φ; thus Θ and Φ are order-preserving bijections and therefore preserve all meets and joins.
To see that Φ preserves multiplication of ideals (the proof for Θ is analogous), we have to demonstrate for I 1 , I 2 ∈ Id(S) that Φ(I 1 )Φ(I 2 ) = Φ(I 1 I 2 ), or equivalently, The set ϕ (QI 1 ⊗ P)ϕ (QI 2 ⊗ P) consists of all finite sums of elements of the form where p 1 , p 2 ∈ P, q 1 , q 2 ∈ Q, s 1 ∈ I 1 and For the opposite inclusion, the set ϕ (QI 1 I 2 ⊗ P) consists of all finite sums of elements of the form ϕ (qs 1 s 2 ⊗ p), where p ∈ P, q ∈ Q, s 1 ∈ I 1 and s 2 ∈ I 2 .Let u ∈ S be chosen such that us 2 = s 2 , and let u = θ (p ′ ⊗ q ′ ).Now applying (1) in reverse gives

Now let I = ∑ m
i=1 Sa i S be a finitely generated ideal.Using the existence of weak local units, let a i = u i a i v i for some u i , v i ∈ S. Using surjectivity of θ , let The opposite inclusion also holds, since for t,t ′ ∈ T , tϕ is finitely generated.
Next, we consider congruences.The following result is the analogue of Theorem 6 in [13] and Theorem 2.15 in [10].However, we give a slightly different proof, which does not need the use of transitive closure.
Theorem 2. If two semirings S and T have common joint weak local units and there exists a unitary Morita context (S, T, S P T , T Q S , θ , ϕ ) with θ , ϕ surjective, then there exists a lattice isomorphism Θ : Con(S) → Con(T ).Furthermore, for each σ ∈ Con(S), S/σ and T /Θ(σ ) are themselves contained in a unitary Morita context with surjective mappings.
Proof.For σ ∈ Con(S), define Clearly Θ(σ ) is an equivalence relation.It is actually a congruence: for (t 1 ,t 2 ), (t The map Θ : Con(S) → Con(T ) is easily seen to be order-preserving.Let Φ be analogous to Θ in the opposite direction: It remains to show that Φ is the inverse of Θ. Due to symmetry, it suffices to prove that ΦΘ = 1 Con(S) .Let σ ∈ Con(S) and s ∼ Φ(Θ(σ )) s ′ .From the definition of Φ, for all p ∈ P and q ∈ Q and from that and the definition of Θ, for all p, p ′ ∈ P and q, q ′ ∈ Q The left side of (2) transforms to Simplifying the right side of (2) in the same way, we get Since σ is compatible with addition and P ⊗ Q consists of finite sums of elements of the form p ⊗ q, we get and from the surjectivity of θ ∀s 1 , s 2 ∈ S : Taking the common joint weak local units for s and s ′ as the values of s 1 and s 2 , we get s ∼ σ s ′ .We have shown that Φ(Θ(σ )) ⊆ σ .In the opposite direction, s or, applying the previously used transformation in reverse, which is equivalent to s ∼ Φ(Θ(σ )) s ′ .Thus Φ(Θ(σ )) = σ , concluding the proof that the congruence lattices are isomorphic.Let τ = Θ(σ ).We proceed to construct a Morita context for S/σ and T /τ.Let µ be the bisemimodule congruence on S P T generated by the set µ 0 of all pairs (sp, s ′ p) and (pt, pt ′ ) where (s, s ′ ) ∈ σ , (t,t ′ ) ∈ τ and p ∈ P. Multiplications S/σ × P/µ → P/µ and P/µ × T /τ → P/µ, (s/σ )(p/µ) := (sp)/µ, (p/µ)(t/τ) := (pt)/µ, are well defined.Now P/µ can be verified to be a unitary (S/σ , T /τ)-module.From now on, we write P/µ to mean S/σ (P/µ) T /τ .
By the above, the map θ : P/µ × Q/ν → S/σ , θ (p/µ, q/ν) := θ0 (p, q) = θ (p ⊗ q)/σ , is well defined.This extends to a monoid homomorphism from the free monoid F(P/µ × Q/ν) to S/σ , which we also denote by θ .We can easily verify that the ordered pairs generating the congruence ρ given in Definition 10 are contained in Ker( θ ).Thus θ factors through ρ, giving a monoid homomorphism θ : The surjectivity of θ implies that θ is also surjective.Now we verify that θ is an (S/σ , S/σ )-bisemimodule homomorphism.Due to additivity, it suffices to consider the tensor product's generators, and due to symmetry, to verify multiplication from the left: By analogy, we get a surjective (T /τ, T /τ)-bisemimodule homomorphism φ : It remains to verify the Morita equations.Due to symmetry, it is enough to verify just one of them.As above, it suffices to consider the tensor product's generators: Thus (S/σ , T /τ, P/µ, Q/ν, θ , φ ) is a unitary Morita context with surjective mappings.

CONCLUSIONS
It seems likely that the existence of a unitary Morita context with surjective mappings would imply Morita equivalence for semirings with local units, as is the case for semirings with identity, and for semigroups and rings with local units.Verifying this is left for future research.If it is true, our results imply that the quantale of ideals and the lattice of congruences are Morita invariants for semirings with suitable local unit conditions.

Definition 3 .Definition 4 .
A bisemimodule over semirings R and S is an algebra R M S = (M, +, 0 M , (r•)| r∈R , (•s)| s∈S ) such that R M is a left R-semimodule, M S is a right S-semimodule, and (rm)s = r(ms) for all r ∈ R, m ∈ M, s ∈ S. Let S be a semiring and S M a left semimodule.For A ⊆ S and U ⊆ M, we define AU = { n ∑ i=1 s i m i : n ∈ N 0 , s i ∈ A, m i ∈ U } and analogously for right semimodules.