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


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 [9-11]. 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 Marki in [13] and for semirings with identity by Sardar and Gupta in [10].


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 [.sub.S]M = (M, +, 0, [(s*)|.sub.s[member of]S]) such that (M, +,0) is a commutative monoid and the following identities hold for all s, s' [member of] S, m, m'< [member of] M:

1.s(m + m')=sm + sm',

2.(s + s')m = sm + s'm,

3.(ss')m = s(s'm),

4.s[0.sub.M] = [0.sub.M],

5.[0.sub.S]m = [0.sub.M].

Right semimodules are defined analogously.

Definition 3. A bisemimodule over semirings R and S is an algebra [.sub.R][M.sub.S] = (M, +,[0.sub.M], [(r*)|.sub.r[member of]R], [(*s)|.sub.s[member of]S]) such that [.sub.R]M is a left R-semimodule, [M.sub.S] is a right S-semimodule, and (rm)s = r(ms) for all r [member of] R, m [member of] M, s [member of] S.

Definition 4. Let S be a semiring and [.sub.S]M a left semimodule. For A [[subset].bar] S and U [[subset].bar] M, we define

[mathematical expression not reproducible]

and analogously for right semimodules.

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.

Definition5. For a semiringS, a left (right) S-semimodule M is unitary if SM = M (MS = M). For semirings S and T, a bisemimodule [.sub.S][M.sub.T] is unitary if [.sub.S]M and [M.sub.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 [member of] S there exist e,e' [member of] S with es = s = se'.

Definition 7. A semiring S has common joint weak local units if for every s,s' [member of] S there exist e,e' [member of] S with s = ese' and s' = es'e'.

Definition 8. An ideal of a semiring S is a set I [[subset].bar] S that is a submonoid of (S, +) and for which SI [[subset].bar] I and IS [[subset].bar] 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 MS and SN, their tensor product M[cross product]N is defined as the factor semigroup of the free commutative additive semigroup F=F(M x N) generated by the set M x N, factorized by the congruence [rho] generated by all ordered pairs of the form

((m + m',n),(m,n) + (m',n)), ((m,n + n'),(m,n) + (m,n')), ((ms,n),(m,sn)),

where m,m' [member of] M,n,n' [member of] N,s [member of] S. The congruence class containing a generator (m,n) of F is denoted by m[cross product]n.

Note that the elements m [cross product] n form a system of generators for the semigroup M [cross product] N, i.e. every element of M [cross product] N is a finite sum of such elements. From the generating pairs of [rho] we obtain the following basic identities:

(m + m') [cross product] n = m [cross product] n + m' [cross product] n, m [cross product] (n + n')=m [cross product] n + m [cross product] n', ms [cross product] n = m [cross product] sn.

The semigroup M [cross product] N is actually a monoid, the zero element being [0.sub.M] [cross product] [0.sub.N]:

m [cross product] n + [0.sub.M] [cross product] [0.sub.N] = m [cross product] n + [0.sub.M] [cross product] [0.sub.S]n = m [cross product] n + [0.sub.M][0.sub.S] [cross product] n = (m + [0.sub.M]) [cross product] n = m [cross product] n.

The tensorproductofsemimodules 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 [.sub.S][M.sub.R] and [.sub.R][N.sub.T] bisemimodules. Then the monoid [M.sub.R] [cross product] [.sub.R]N can be turned in a unique way into a bisemimodule [.sub.S][M.sub.R] [cross product] [.sub.R][N.sub.T], retaining its addition and zero element, such that for any m [member of] M,n [member of] N

s(m [cross product] n)=sm [cross product] n, (m [cross product] n)t = m [cross product] nt.

Definition 11. A Morita context is a sextuple (S, T,[.sub.S][P.sub.T],[.sub.T][Q.sub.S], [theta],[phi]) where

1. S and T are semirings;

2. [.sub.S][P.sub.T] and [.sub.T][Q.sub.S] are bisemimodules as indicated by the subscripts;

3. [theta]: [.sub.S][(P [cross product] Q).sub.S] [right arrow] [.sub.S][S.sub.S] and [phi]: [.sub.T][(Q[cross product]P).sub.T] [right arrow] [.sub.T][T.sub.T] are bisemimodule homomorphisms; 4. for every p, p' [member of] P,[theta](p [cross product] q)p' = p[phi](q [cross product] p');

5. for every q,q' [member of] Q, [phi](q [cross product] p)q' = q[theta](p [cross product] q').

We say that a Morita context (S, T,[.sub.S][P.sub.T],[.sub.T][Q.sub.S],[theta],[phi]) is unitary if [.sub.S][P.sub.T] and [.sub.T][Q.sub.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 [rho] be the congruence on F generated by the pair (y,[y.sup.2]), and let R := F/[rho]. Then e := y/[rho] is an idempotent. Let S be the subsemiring ReR of R; then S = SeS [not equal to] eSe. Now one can verify that (S,eSe,[.sub.S][Se.sub.eSe,eSe][eS.sub.S],[theta] ,[phi]), where [theta] (se[cross product]es') := ses' and [phi](es[cross product]s'e) := ess'e, is aunitary Morita context with surjective mappings.


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,[.sub.S][P.sub.T,T][Q.sub.S],[theta],[phi]) with [theta],[phi] surjective, then there is a quantale isomorphism Id(S) [right arrow] Id(T) that takes finitely generated ideals to finitely generated ideals.

Proof. Let(S,T,[.sub.S][P.sub.T,T][Q.sub.S],[theta],[phi]) beaunitary Morita context with [theta], [phi] surjective. Define

[mathematical expression not reproducible],

[mathematical expression not reproducible],

It is easily seen that the sets on the right side are indeed ideals. We show that [THETA] and [PHI] are mutually inverse bijections. Due to symmetry, it suffices to show that [THETA]([PHI](I)) = I for any I [member of] Id(S). First,

[THETA]([PHI](I)) = [theta](P[phi](QI[cross product]P)[cross product]Q).

We now show that

[theta](P[phi](QI[cross product]P)[cross product]Q) = [theta](P[cross product]Q)I[theta](P[cross product]Q).

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 [theta](p[phi](qs[cross product]p')[cross product]q'), where s [member of] I, p,p' [member of] P, q,q' [member of] Q. This transforms into

[theta](p[phi](qs[cross product]p') [cross product]q') = [theta]([theta] (p[cross product]qs)p'[cross product]q') = [theta] (p[cross product]q)s[theta](p'[cross product]q'),

and elements of this form generate the set on the right side. Now [theta](P[cross product]Q)I[theta](P[cross product]Q) = SIS [[subset].bar] I. Using weak local units, we can see that I [[subset].bar] SIS, concluding the proof that [THETA] and [PHI] are mutually inverse.

It is easy to see that for J' [[subset].bar] J, [THETA](J) [[subset].bar] [THETA](J) and the same for [PHI]; thus [THETA] and [PHI] are order-preserving bijections and therefore preserve all meets and joins.

To see that [PHI] preserves multiplication of ideals (the proof for [THETA] is analogous), we have to demonstrate for [I.sub.1],[I.sub.2] [member of] Id(S) that [PHI]([I21])[PHI]([I.sub.2]) = [PHI]([I.sub.1] [I.sub.2]), or equivalently,

[phi]([QI.sub.1][cross product]P)[phi]([QI.sub.2][cross product]P) = [phi]([QI.sub.1][I.sub.2] [cross product] P).

The set [phi]([QI.sub.1] [cross product] P)[phi]([QI.sub.2] [cross product] P) consists of all finite sums of elements of the form

[phi]([q.sub.1][s.sub.1] [cross product] [p.sub.1])[phi]([q.sub.2][s.sub.2][cross product][p.sub.2]) = [phi]([q.sub.1][s.sub.1][cross product][p.sub.1][phi]([q.sub.2][s.sub.2][cross product][p.sub.2])) = [phi]([q.sub.1][s.sub.1][cross product][theta]([p.sub.1][cross product][q.sub.2][s.sub.2])[p.sub.2]) = [phi]([q.sub.1][s.sub.1][theta]([p.sub.1][cross product][q.sub.2])[s.sub.2][cross product][p.sub.2]), (1)

where [p.sub.1],[p.sub.2] [member of] P, [q.sub.1],[q.sub.2] [member of] Q, [s.sub.1] [member of] [I.sub.1] and [s.sub.2] [member of] [I.sub.2]. Since [s.sub.1][theta]([p.sub.1] [cross product] [q.sub.2])[s.sub.2] [member of] [I.sub.1] [I.sub.2], we have shown [PHI]([I.sub.1])[PHI]([I.sub.2]) [[subset].bar] [PHI]([I.sub.1][I.sub.2]).

For the opposite inclusion, the set [phi]([QI.sub.1] [I.sub.2] [cross product] P) consists of all finite sums of elements of the form [phi]([qs.sub.1][s.sub.2][cross product]p), where p [member of] P, q [member of] Q, [s.sub.1] [member of] [I.sub.1] and [s.sub.2] [member of] [I.sub.2]. Let u [member of] S be chosen such that [us.sub.2] = [s.sub.2], and let u = [theta](p' [cross product] q'). Now applying (1) in reverse gives

[phi]([qs.sub.1][s.sub.2][cross product]p)=[phi]([qs.sub.1][theta](p' [cross product] q') [s.sub.2] [cross product] p) = [phi]([qs.sub.1] [cross product] p')[phi](q'[s.sub.2] [cross product] p) [member of] [phi] ([QI.sub.1] [cross product] P) [phi] ([QI.sub.2] [cross product] P).

Now let [mathematical expression not reproducible] [Sa.sub.i]S be a finitely generated ideal. Using the existence of weak local units, let [a.sub.i] = [u.sub.i][a.sub.i][v.sub.i] for some [u.sub.i], [v.sub.i] [member of] S. Using surjectivity of [theta], let [mathematical expression not reproducible]. Now

[mathematical expression not reproducible].

The opposite inclusion also holds, since for [mathematical expression not reproducible]. Therefore

[mathematical expression not reproducible]

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,[.sub.S]PT,[.sub.T]QS,[theta],[phi]) with [theta],[phi] surjective, then there exists a lattice isomorphism [THETA] : Con(S) [right arrow] Con(T). Furthermore, for each [sigma] [member of] Con(S), S/[sigma] and T/[THETA]([sigma]) are themselves contained in a unitary Morita context with surjective mappings.

Proof For [sigma] [member of] Con(S), define

[mathematical expression not reproducible].

Clearly [THETA]([sigma]) is an equivalence relation. It is actually a congruence: for ([t.sub.1], [t.sub.2]),([t.sub.3],[t.sub.4]) [member of] [THETA]([sigma]),p [member of] P,q [member of] Q

[mathematical expression not reproducible]


[mathematical expression not reproducible]

The map [THETA]: Con(S) [right arrow] Con(T) is easily seen to be order-preserving. Let [PHI] be analogous to [THETA] in the opposite direction:

[mathematical expression not reproducible]

It remains to show that [PHI] is the inverse of [THETA]. Due to symmetry, it suffices to prove that [PHI][THETA] = [1.sub.Con(s)]. Let [sigma] [member of] Con(S) and [mathematical expression not reproducible]. From the definition of [PHI], for all p [member of] P and q [member of] Q

[mathematical expression not reproducible]

and from that and the definition of [THETA], for all p, p' [member of] P and q,q' [member of] Q

[mathematical expression not reproducible] (2)

The left side of (2) transforms to

[mathematical expression not reproducible]

Simplifying the right side of (2) in the same way, we get

[mathematical expression not reproducible]

Since [sigma] is compatible with addition and P [cross product] Q consists of finite sums of elements of the form p [cross product] q, we get

[mathematical expression not reproducible]


[mathematical expression not reproducible]

Taking the common joint weak local units for s and s' as the values of [s.sub.1] and [s.sub.2], we get [mathematical expression not reproducible]. We have shown that [PHI]([THETA]([sigma])) [[subset].bar] [sigma].

In the opposite direction, [mathematical expression not reproducible] implies

[mathematical expression not reproducible]

or, applying the previously used transformation in reverse,

[mathematical expression not reproducible]

which is equivalent to [mathematical expression not reproducible]. Thus [PHI]([THETA]([sigma])) = [sigma], concluding the proof that the congruence lattices are isomorphic.

Let [tau] = [THETA]([sigma]). We proceed to construct a Morita context for S/[sigma] and T/[tau].

Let [mu] be the bisemimodule congruence on [.sub.S][P.sub.T] generated by the set [[mu].sub.0] of all pairs (sp,s'p) and (pt, pt') where (s, s') [member of] [sigma], (t ,t') [member of] [tau] and p [member of] P. Multiplications S/[sigma] x P/[mu] [right arrow] P/[mu] and P/[mu] x T/[tau] [right arrow] P/[mu],

(s/[sigma])(p/[mu]) := (sp)/[mu], (p/[mu])(t / [tau]) := (pt)/[mu],

are well defined. Now P/[mu] can be verified to be a unitary (S/[sigma], T/[tau])-module. From now on, we write P/[mu] tomean [.sub.S/[sigma]][(P/[mu]).sub.T/[tau]].

Analogously, we define v [member of] Con([.sub.T][Q.sub.S]) generated by the set of pairs [v.sub.0], and Q/v becomes a unitary (T/[tau],S/[sigma])-bisemimodule.

Denote by ([mu], v) the equivalence relation {((p,q), (p',q')) : (p,p') [member of] [mu], (q,q) [member of] v} on the set P x Q. Defineamap [[??].sub.0]: P x Q [right arrow] S/[sigma] by

[mathematical expression not reproducible].

We now verify that ([mu], v) [[subset].bar] Ker([[??].sub.0]). Clearly, ([mu], v) is generated by the set

[mathematical expression not reproducible]

and it suffices to show that this set is contained in Ker([[??].sub.0]). Consider the case of ([p.sub.1],[p.sub.2]) [member of] [[mu].sub.0],q [member of] Q (the case of ([q.sub.1], [q.sub.2]) [member of] [v.sub.0], p [member of] P is analogous). There are two possibilities.

(a) ([p.sub.1], [p.sub.2]) = (sp, s'p) for [mathematical expression not reproducible], p [member of] P. Then

[mathematical expression not reproducible]

and thns [mathematical expression not reproducible]

(b) [mathematical expression not reproducible]. Then the definition of [tau] = [THETA]([sigma]) implies that [theta](pt[cross product] q) ~[.sub.[sigma]] [theta](pt'[cross product]q), and thus [mathematical expression not reproducible].

By the above, the map [mathematical expression not reproducible],

[mathematical expression not reproducible],

is well defined. This extends to a monoid homomorphism from the free monoid F(P/[mu] x Q/v) to S/[sigma], which we also denote by [theta]. We can easily verify that the ordered pairs generating the congruence [rho] given in Definition 10 are contained in Ker([??]). Thus [theta] factors through [rho], giving a monoid homomorphism [??] :P/[mu][cross product]Q/v [right arrow] S/[sigma],

[mathematical expression not reproducible]

The surjectivity of [theta] implies that [??] is also surjective.

Now we verify that [??] is an (S/[sigma], S/[sigma])-bisemimodule homomorphism. Due to additivity, it suffices to consider the tensor product's generators, and due to symmetry, to verify multiplication from the left:

[mathematical expression not reproducible]

By analogy, we get a surjective (T/[tau], T/[tau])-bisemimodule homomorphism [??] : Q/v [cross product] P/[mu] [right arrow] T/[tau],

[mathematical expression not reproducible]

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:

[mathematical expression not reproducible]

Thus (S/[sigma], T/[tau],P/[mu], Q/v, [??], [ ??]) is a unitary Morita context with surjective mappings. []


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.


The author thanks his doctoral advisor Valdis Laan, who gave several ideas and helped greatly in proofreading the manuscript. The research was partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The publication costs of this article were covered by the Estonian Academy of Sciences.


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Laur Tooming

Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2, 50409 Tartu, Estonia;

Received 5 October 2017, revised 16 February 2018, accepted 19 February 2018, available online 20 June 2018

Lokaalsete uhikutega poolringide Morita kontekstid ja ideaalid ning kongruentsid

Laur Tooming

Klassikaliselt [1;2, ptk 6] on Morita ekvivalentsus defineeritud uhikelemendiga ringide klassil: kaht ringi peetakse ekvivalentseks, kui nende vasakpoolsete (ja samavaarselt parempoolsete) moodulite kategooriad on ekvivalentsed. See tingimus on samavaarne nn Morita konteksti olemasoluga. Morita kontekst koosneb kahest bimoodulist ule nende kahe ringi ja kahest homomorfismist nende bimoodulite tensorkorrutistest ringidesse, mis peavad rahuldama teatud tingimusi. Morita konteksti definitsioon voib tunduda keeruline, aga selle abil on mitmeid vaiteid Morita ekvivalentsuse kohta toestada lihtsam kui kategoorse definitsiooni abil.

Morita ekvivalentsuse moistet on mitmel viisil edukalt uldistatud muudele struktuuridele kui uhikelemendiga ringid. Esiteks on vaadeldud mitmesuguste lokaalsete uhikutega ringe [3,4]. Sel juhul tuleb Morita ekvivalentsuse ja Morita konteksti definitsioonides asendada suvalised moodulid unitaarsete moodulitega.

Teine uldistussuund on olnud ringide ja moodulite asendamine poolruhmade ning polugoonidega. Monoidide korral [5,6] on Morita ekvivalentsus vaga lahendane isomorfismile ja seetottu ei paku eriti huvi. Kui aga nouda (analoogiliselt ringidega) poolruhmadelt uhikelemendi asemel lokaalsete uhikute olemasolu ja vaadelda suvaliste polugoonide asemel unitaarseid, tekib sisukas teooria, kus kehtivad mitmed ringidega analoogilised tulemused [7].

Poolring on algebraline struktuur, kus Abeli ruhma aditiivne struktuur ringi definitsioonis on asendatud monoidiga. Ringi moodulitele vastavad poolringi poolmoodulid. On loomulik kusimus, kas Morita teooriat on voimalik arendada ka poolringide korral ja kas see on lahedasem ringide voi poolruhmade Morita teooriale. Uhikelemendiga poolringide jaoks uurisid Morita ekvivalentsust esimestena Katsov ja Nam [8] ning edasi Sardar, Gupta ja Saha [9-11]. Lokaalsete uhikutega poolringide jaoks uuris Morita ekvivalentsust esimesena Liu [12].

Kaesolevas artiklis laheneme lokaalsete uhikutega poolringide Morita teooriale teisest suunast--Morita kontekstide poolt. Sel juhul ei ole veel toestatud Morita ekvivalentsuse samavaarsus Morita konteksti olemasoluga. Naitame, et Morita konteksti leidumisest koos tingimustega, mis on analoogilised lokaalsete uhikutega ringide ja poolruhmade jaoks vajalike tingimustega, jareldub, et kahel poolringil on isomorfsed ideaalide ning kongruentside vored. Analoogilisi tulemusi on varem poolruhmade jaoks saanud Laan ja Marki [13] ning uhikelemendiga poolringide jaoks Sardar ja Gupta [10].
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Author:Tooming, Laur
Publication:Proceedings of the Estonian Academy of Sciences
Article Type:Report
Date:Sep 1, 2018
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