Abstract
In this paper we characterize the class of semirings S for which the semirings of square matrices Mn(S) over S are (left) k‐artinian. Also an analogue of the Hilbert basis theorem for semirings is obtained.Corrigendum to “Chain conditions on semirings”dx.doi.org/10.1155/S0161171297000550
Highlights
INTRODUCTION A semiringS is defined as an algebraic system (S,+,.) such that (S,+) and (S,) are semigroups, connected by a(b + c) ab + ac and (b + c)a ba + ca for all a, b, c e S
A semiring S is defined as an algebraic system (S,+,.) such that (S,+) and (S,) are semigroups, connected by a(b + c) ab + ac and (b + c)a ba + ca for all a, b, c e S
A semifield is a semiring in which non-zero elements form a group under multiplication
Summary
INTRODUCTION A semiringS is defined as an algebraic system (S,+,.) such that (S,+) and (S,-) are semigroups, connected by a(b + c) ab + ac and (b + c)a ba + ca for all a, b, c e S. Examples show that there are (left) artinian semirings (even semifields) S for which Mn(S) is not (left) In this paper we characterize the class of semirings S such that all Mn(S) are (left) k-artinian (cf Definition 1.1). Another characterization of the class of semirings S for which all Mn(S) are (left) h-artinian is obtained.
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