MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY TRN GIANG NAM MORITA EQUIVALENCE FOR SEMIRINGS AND CHARACTERIZE SOME CLASSES OF SEMIRINGS ABSTRACT OF THE DOCTORAL THESIS Speciality: Algebra and theory of numbers Code: 62.46.05.01 Supervisors 1. ASSOC. PROF. SCI. DR. NGUYEN XUAN TUYEN 2. ASSOC. PROF. DR. NGO SY TUNG VINH - 2011 1 Introduction Semiring is introduced by Vandiver in 1934, generalize the notion of non- commutative rings in the sense that negative elements don’t have exist. In this thesis, semirings were assumed to have both additive identity and multiplicative identiy. Nowadays one may clearly notice a growing interest in developing the alge- braic theory of semirings and their numerous connections with, and applications in, different branches of mathematics, computer science, quantum physics, and many other areas of science (see, for example, the recently published Glazek (2002)). As algebraic objects, semirings certainly are the most natural gen- eralization of such (at first glance different) algebraic systems as rings and bounded distributive lattices. As is well known, structure theories for alge- bras of classes/varieties of algebras constitute an important “classical” area of the sustained interest in algebraic research. In such theories, so-called simple algebras — algebras possessing only the identity and universal congruences — play a very important role of “building blocks.” In contrast to the varieties of groups and rings, research on simple semirings has been just recently started and, therefore, not much on the subject is known (for some recent results on this subject one may consult Bashir et al. (2001), Monico (2004), Bashir-Kepka (2007) and Zumbragel (2008)). In the same time in the modern homological theory of semimodules over semirings, the results characterizing semirings by properties of semimodules and/or suitable categories of semimodules over them are of great importance and sustained interest (for some recent such results one may consult, for example, Il’in and Katsov (2004), S. N. Il’in (2008), Y. Katsov (2004), O. Sokratova (2002)). At any rate, this thesis concerns the ideal- and congruence-simpleness- in a semiring setting, these two notions of simpleness are not the same and should be differed-for some classes of semirings. Also, comfirming a conjecture and solving a problem of Katsov in the class of additively regular semisimple semirings. The thesis is organized as follows. In Chaper 1, we present the number of new results on projective semimodules, which extend to a non-additive semiring setting the corresponding classical important facts on projective modules over 2 rings. We develop from two different, eventually equivalent, perpectives the “Morita equibalence” concept in the category of semirings. In Chapter 2, we establish that ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariants for semirings; and describe all simple semirings having projective minimal left (right) ideals, and provide a characterization of ideal-simple semirings having projective minimal left (right) ideals. In Chapter 3, we describe all ideal-simple and simple artinian aic-semirings, as well as congruence-simple lattice-orderd semirings. Further, we also present a complete desciription of left (right) subtractive semisimple semirings. Then, apply this result, we describe left (right) subtractive artinian ideal-simple semir- ings, as well as left (right) subtractive artinian congruence-simple semirings. Finally, we describe all simple semirings with an infinite element, and provide a characterization of ideal-simple semirings with an elements. In Chapter 4, we describe all additively idempotent semisimple semirings over that the notions of either projectiveness and flatness, or flatness and mono- flatness for semimodules coincide. Also, we characterize semisimple semirings by projective and injective semimodules. 3 Chapter 1 MORITA EQUIVALENCE 1.1 Background In this section, we recall notions and facts on semirings and semimodules over them that can be found in Golan (1999). Also, we recall the notion of tensor product for semimodules in sense of Y. Katsov (1997). From now on, let M R and R M be the categories of right and left semimodules, respectively, over a semiring R. 1.2 Progenerators In this section, we characterize, and describe progenerators — finitely generated projective generators — of semimodule categories. Now let S = R M(P, P ) := End ( R P ) be a semiring of all endomorphisms of a left R-semimodule R P ∈ | R M|. Then, considering endomorphisms of R P operating on the right of R P , one easily sees that P becomes an R-S- semimodule, i.e., R P S ∈ | R M S |. As in the classical case of modules over rings, we write Q = P ∗ := R M( R P, R R) for the dual S-R-semimodule P ∗ of the R- S-semimodule P. We also define the endomorphism qp ∈ S by p  (qp) = (p  q)p for any p ∈ P and q ∈ Q. Then, in the same fashion as has been done in the classical case of modules over rings, one can easily obtain the following. Lemma 1.2.2. The assignments (p, q) −→ pq and (q, p) −→ qp define the (R, R)-homomorphism α : P ⊗ S Q −→ R and the (S, S)-homomorphism 4 β : Q ⊗ R P −→ S, respectively. Our next observation provides a characterization of finitely generated pro- jective left R-semimodules R P ∈ | R M| by means of the homomorphism β : Q ⊗ R P = P ∗ ⊗ R P −→ S = End ( R P ) from Lemma 1.2.2 above. Proposition 1.2.4. A left R-semimodules R P ∈ | R M| is finitely generated and projective iff β : Q ⊗ R P −→ S is a surjection. Now we turn to generators in the semimodule categories. Definition 1.2.6. A left semimodule R P ∈ | R M| is said to be a generator for the category of left semimodules R M if the regular semimodule R R ∈ | R M| is a retract of a finite direct sum ⊕ i P of the semimodule R P . Similar to Propositions 1.2.4, our next observation provides a characterization of finitely generated generator R P ∈ | R M| for R M by means of the homomor- phism the (R, R)-homomorphism α : P ⊗ S Q −→ R from Lemma 1.2.2. above. Proposition 1.2.8. A finitely generated left semimodule R P ∈ | R M| is a generator for R M iff the (R, R)-homomorphism α : P ⊗ S Q −→ R is a surjection. Moreover, if α is a surjection, then it is an isomorphism. Combining the concepts of ‘generator’ and ‘finitely generated projective semimodule’, we come up with a new concept of ‘progenerator’, namely: A left semimodule R P ∈ | R M| is said to be a progenerator for the category of left semimodules R M if it is a finitely generated projective generator. Then, from Propositions 1.2.4 and 1.2.8, one obtains the following important result. Theorem 1.2.9. A left R-semimodules R P ∈ | R M| is a progenerator iff the homomorphisms α : P ⊗ S Q −→ R and β : Q ⊗ R P −→ S are bisemimodule isomorphisms. 5 1.3 Morita Equivalence The concept of Morita equivalence for semirings, or on the category SRing of semirings, can be introduced in different fashions. And for our goals we have found that following approach to be good one. Definition 1.3.1. Semirings R and S is said to be Morita equivalent if there exists a progenerator R P ∈ | R M| for R M such that the semirings S and End ( R P ) are isomorphic, i.e., S ∼ = End ( R P ) as semirings. To present an alternative approach to the concept of Morita equivalence, we need the following result that is is a nonadditive analog of the Eilenberg-Watts theorem for module categories in the semimodule setting. Theorem 1.3.5. For a functor F : M R −→ M S the following statements are equivalent: (i) F has a right adjoint; (ii) F is right continuos and preserves coproducts (direct sums); (iii) There exists unique up to natural isomorphism a R-S-bisemimodules P ∈ | R M S | such that the functors − ⊗ R P : M R −→ M S and F are naturally isomorphic, i.e., F ∼ = − ⊗ R P . Using Theorem 1.3.5, one gets the following important and intriguing con- sequence of the Morita equivalence between two semirings R and S is the fact that reasonable important corresponding categories of semimodules over these semirings are equivalent, as categories, as well. Theorem 1.3.12. For semirings R and S the following conditions are equiva- lent: (i) The semirings R and S are Morita equivalent semirings; (ii) The semimodule categories M R and M S are equivalent categories; (iii) The semimodule categories R M and S M are equivalent categories. 6 1.4 Conclusion In this chapter, establishing a sufficient and necessary condition to a finitely generated semimodule is either projective (Propostion 1.2.4), or a generator (Proposition 1.2.8). Characterizing properties of progenerators of semimodule categories (Theorem 1.2.9). Also, we develop from two different, eventually equivalent, perspectives the ‘Morita equivalence’ concept in the nonadditive setting of the category of semirings. Among several, as we hope useful for the future research, observations obtained in this chapter, there are two central results — describing all covariant functors having right adjoints (Theorem 1.3.5), and characterizing Morita equivalence for semirings via semimodule categories (Theorem 4.12). 7 Chapter 2 MORITA INVARIANT AND ITS APPLICATIONS 2.1 Endomorphism semirings A semiring R is congruence-simple if the diagonal, id R , and universal, R 2 , congruences are the only congruences on R; and R is ideal-simple if 0 and R are its only ideals. A semiring R is said to be simple if it is simultaneously congruence- and ideal-simple. Congruence-simple and ideal-simple semirings have been studied by C. Mon- ico (2004), J. Zumbragel (2008), J. Jeˇzek - T. Kepka - M. Mar´oti (2009), Bourne- Zassenhaus (1957), O. Steinfeld - R. Wiegandt (1967), Stone (1977), Weinert (1984), A surjective homomorphism of semirings f : R −→ S is called strongly semiisomorphic if Ker(f) := f −1 (0) = {0} and f(I)  S for any proper ideal I of R. A semiring R is called additively idempotent if the monoid (R, +) is additively idempotent. Our next proposition illustrates that ideal-simple semirings can be studied by additively idempotent simple semirings. Proposition 2.1.6. A semirng R is ideal-simple iff R is a simple ring, or there exists a strong semiisomorphism from R onto an additively idempotent, simple semiring S. 8 In the light of Proposition 2.1.6, it is natural to bring up that ideal- simple (simple) semirings can be understood by simpleness of subsemirings of endomorphism semirings of idempotent commutative monoids. Our next result describes simple endomorphism semirings of idempotent commutative monoids. Theorem 1.2.9. The following conditions for the endomorphism hemiring End(M) of an idempotent commutative monoid (M, +, 0) are equivalent: (i) End(M) is simple; (ii) End(M) is ideal-simple; (iii) The semilattice M is a finite distributive lattice. 2.2 Morita invariants Our next result establishes that ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariants for semirings. Theorem 2.2.6. Let R and S be semirings. If R is Morita equivalent to S, then R congruence-simple (ideal-simple) iff S is congruence-simple (ideal-simple); in particular, R is simple iff S is simple. The following result shows that direct limits preverses ideal-simpleness, congruence-simpleness and simpleness of semirings. Proposition 2.2.7. Let {R i | R i ∈ |SR|, i ∈ I} be a directed family of semirings and R = lim −→ I R i . If R i , i ∈ I, are (i) ideal-simple, (ii) congruence-simple, or (iii) left (right) subtractive, then R is ideal-simple, congruence-simple, or left (right) subtractive, respectively, too. Now let us consider an explicit, well known for rings, construction of the direct limit situation in a semiring setting. Fix a semiring D and consider semirings R i = M 2 i (D) (i ≥ 0) of square matrices of order 2 i over D. We shall regard R i as a subsemiring of R i+1 by identifying a 2 i × 2 i matrix M with the 2 i+1 × 2 i+1 9 matrix  M 0 0 M  . In this way, we have a chain of semirings R 0 ⊆ R 1 ⊆ R 2 ⊆ . . . , where R 0 = D; and it is clear that the direct limit R = lim −→ I R i of the directed family {R i | i ∈ I} is, in fact, the union ∪ I R i of the semirings R i , i ∈ I. Proposition 2.2.8. The semiring R = lim −→ I R i is not left artinian. In light of Propositions 2.2.7 and 2.2.8 and taking into consideration that even for rings the structure of general Congruence-simple, ideal-simple and simple rings remains difficult. Therefore, in the next section and chapter, we only study the above semirings for some special classes of semirings. 2.3 Applications Our following observation is an analog of the famous “Double Centralizer Property” of ideals of simple rings in the setting of simple semirings. Theorem 2.3.1. Let R be a simple semiring, and I be a nonzero left ideal. Let D = End ( R I) (viewed as a semiring of right operators on I). Then (i) The natural map f : R −→ End (I D ) is a semiring isomorphism; (ii) I is a generator in the category of semimodule R M, and a finitely generated projective right D− semimodule; (iii) There exists a natural number n and an idempotent e in matrix semiring M n (D) such that R ∼ = eM n (D)e; (iv) D is simple iff I is a finitely generated projective left R-semimodule. As a corollary of Theorem 2.3.1, we obtain the description of all simple semirings having projective minimal left (right) ideals. Theorem 2.3.2. For a semiring R, the followings are equivalent: (i) R is a simple semiring containing a projective minimal left ideal; (ii) R is a simple semiring containing a projective minimal right ideal; [...]... semisimple semirings by projective and injective semimodules 19 Conclusions of this thesis In this thesis, one obtains the following: 1 Describe Morita equivalence for semirings Establish that congruence- simpleness, ideal-simpleness and simpleness of semirings are Morita invariant 2 Describe simple semirings containing a projective minimal one-sided ideal Rresent ideal-simple semirings containing a projective... semirings with an infinite element and comple simple semirings Theorem 3.3.9 For a semiring R, the following conditions are equivalent: (i) R is a simple semiring containing an infinite element; (ii) R is Morita equivalent to the Boolean semiring B; (iii) R ∼ EM , where M is a nonzero finite distributive lattice; = (iv) R is a complete simple semiring Combining Proposition 2.1.6 and Theorem 3.3.9, we immediately... 2.1.6), and describing all simple endomorphism semirings of idempotent commutative monoids (Theorem 2.1.9) Also, establishing that ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariants for semirings (Theorem 2.2.6) Describing all simple semirings having projective minimal left (right) ideals (Theorem 2.3.2), and provide a characterization of ideal-simple semirings having... Colloquium, 16, 415-426 2 Y Katsov, T G Nam, N X Tuyen (2011), More on Subtractive Semirings: Simpleness, Perfectness and Related Problems, Communications in Algebra (accepted) 3 Y Katsov and T G Nam (2011), Morita Equivalence and Homological Characterization of Semirings, Journal of Algebra and Its Applications, 10, 449–473 4 Y Katsov, T G Nam and J Zumbragel, On Simpleness of Semirings and Complete Semirings . perpectives the Morita equibalence” concept in the category of semirings. In Chapter 2, we establish that ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariants. the homomorphisms α : P ⊗ S Q −→ R and β : Q ⊗ R P −→ S are bisemimodule isomorphisms. 5 1.3 Morita Equivalence The concept of Morita equivalence for semirings, or on the category SRing of semirings, can be. having right adjoints (Theorem 1.3.5), and characterizing Morita equivalence for semirings via semimodule categories (Theorem 4.12). 7 Chapter 2 MORITA INVARIANT AND ITS APPLICATIONS 2.1 Endomorphism