Toward homological characterization of semirings by e-injective semimodules

Abstract

In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize onesided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.

Homology of systemic modules

“…We can define a specific chain complex from a given module M with negation, applying symmetrization to the classical theory given in [9,35,48]. We define (−) 1 Tensor products over semirings parallel tensor products over rings, and are well studied in the literature [36,37,38]; the systemic version is given in [45, § 6.4], where it is stipulated that (−)(b ⊗ b ′ ) = ((−)b) ⊗ b ′ for all b, b ′ . Given a commutative semiring system (A, T , (−), ) and a systemic module M we define…”

“…The surpassing relation plays a key structural role. 1 In [34] we have considered both "ground systems" which often are semirings, and systemic modules over ground systems. In this paper we investigate the rudiments of homology theory of systemic modules over a given ground system.…”

“…In this paper we study chains of modules and long exact sequences, in terms of the surpassing relation , and provide a systemic version of [1]. One major obstacle in obtaining a meaningful homology theory for semirings is the lack of correspondence between ideals (modules) and kernels of homomorphisms (congruences).…”

“…Here we continue with projective dimension via a version of Schanuel's Lemma which is more technical but which fits in well with the systemic environment, i.e., by means of negation maps and symmetrization as in [3], [45], §2.7. 1 There is a related weaker relation ∇, where a 1 ∇a 2 if and only if 0 a 1 (−)a 2 , which also restricts to equality on T . The relation ∇ is reflexive and symmetric, but not transitive.…”