On Quasi-Frobenius Rings

FP-injective semirings, semigroup rings and Leavitt path algebras

“…(We note that the equivalence of parts (i) and (iv) of Theorem 3.4 has appeared in [16,Theorem 5.41] for the ring case.) Theorem 3.4.…”

“…By noting that a left (respectively, right) subsemimodule X ⊆ S n is finitely generated if and only if X = Row(A) (respectively, X = Col(A)) for some matrix A over S, it is clear that this notion of exactness may be naturally separated into notions of left and right exactness. We begin by comparing left exactness with three related definitions, which are well-known in the context of ring theory (see [16], for example):…”

“…When S is commutative these notions coincide, and we drop the left/right prefix. These properties are not leftright symmetric in general -see [16,Example 5.46] for the case of FP-injectivity.…”

“…Below we give several equivalent characterisations of FP-injective semirings, some of which are the semiring analogues of results for FP-injective rings (see [16,Chapter 5]). …”

“…We recall that a ring is called quasi-Frobenius if it is (left and right) Artinian and self-injective. (There are numerous equivalent conditions characterising quasi-Frobenius rings -see [16] for example). One of the main open problems concerning quasi-Frobenius rings is the so-called Faith-Menal conjecture (see, e.g., [16, pp.…”