FP-injective semirings, semigroup rings and Leavitt path algebras

Abstract

Abstract. In this paper we give characterisations of FP-injective semirings (previously termed "exact" semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FPinjectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.

Which semifields are exact?

“…In particular, we define right im A as the set of all vectors in S d×1 that can be written as Aw with some w ∈ S n×1 . We proceed with a characterization of exactness that looks very similar to Theorem 3.2 in [13] and to Lemma 3.3 in [5]. Therefore, the following result cannot be called 'new', and we provide the proof just for the sake of completeness.…”

“…Therefore, the division rings are the first examples of exact semirings. Let us also point out that, in the case of rings, the exactness is equivalent to the property known as FP-injectivity, see [3,5,12].…”

“…The subsequent paper [13] contains the exactness proofs for other related semirings, including T = (R ∪ {+∞, −∞}, min, +). Another proof that T is exact is contained in the paper [5], where the authors do also prove that an additively cancellative semifield is exact if and only if it is a field.…”