This chapter discusses semirings and semifields. Among many concepts of universal algebras, semirings are those that originate from rings, by cancelling the assumption that ( R , +) has to be a group. Depending on how much other ring-like properties are also cancelled or added, various different concepts of semirings (S, +, ·) have been considered. Semirings with different properties have become important in theoretical computer science. A universal algebra S = (S, +, ·) with a nonempty set S and two binary operations, written as addition and multiplication, is called a “semiring” if (S, +) and (S,·) are arbitrary semigroups, such that a ( b + c ) = ab + ac and ( b + c ) a = ba + ca hold for all a, b, c ∈ S . In particular, ( S , +,·) is called a “proper semiring” if ( S , +) is not a group. Each distributive lattice is a semiring, clearly commutative and idempotent with respect to both operations. It has a zero or an identity if it is bounded from below or above, respectively.

Semirings and Semifields

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.

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Chain Conditions on Semirings

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Matrix representation of simple halfrings

Analogues of Artin-Wedderburn and Goldie structure theorems are obtained for a class of halfrings which includes the semisubtractive ones. In the semisubtractive case, precise results are obtained which show that non-ring examples of these structures are relatively limited.

On semisubtractive halfrings

Ideal types are used to classify ideals in a polynomial halfring and to determine when an ideal behaves like a fc-ideal. In particular, these results are used to classify all ideals in the ring of polynomials over the integers.

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Ideal types in a polynomial halfring

https://www.ams.org/proc/1972-033-01/S0002-9939-1972-0291228-5/S0002-9939-1972-0291228-5.pdf

H. E. Stone

Papers

Ideals in halfrings