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.

Ideals in halfrings

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.

/pdf/chain-conditions-on-semirings-1c9xj7v8op.pdf

Chain Conditions on Semirings

/pdf/matrix-representation-of-simple-halfrings-51x06x6br6.pdf

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.

background

/pdf/ideal-types-in-a-polynomial-halfring-16pb1pwxx7.pdf

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

1. Introduction. When studying ideal theory in semirings, it is natural to consider the quotient structure of a semiring modulo an ideal. If 7 is an ideal in a semiring R, the collection {x+l}xeit oí sets x + I={x+i\iEl} need not be a partition of R. Faced with this problem, [2] used equivalence relations to determine cosets relative to an ideal. La Torre successfully established analogues of several well-known isomorphism theorems for rings. However, the methods that Bourne and La Torre used to construct quotient structures proved to be unsuccessful when trying to obtain an exact analogue of the Fundamental Theorem of Homomorphisms. In this paper, the notion of a Ç-ideal will be defined and a construction process will be presented by which one can build the quotient structure of a semiring modulo a Q-ideal. Maximal homomorphisms will be defined and examples of such homomorphisms will be given. Using these notions, the Fundamental Theorem of Homomorphisms will be generalized to include a large class of semirings.

/pdf/a-fundamental-theorem-of-homomorphisms-for-semirings-latw3nx74i.pdf

Ideals in halfrings

Citations

Semirings and Semifields

Chain Conditions on Semirings

Matrix representation of simple halfrings

On semisubtractive halfrings

Ideal types in a polynomial halfring

References

A fundamental theorem of homomorphisms for semirings

On the Homomorphism Theorem for Semirings.

Related Papers (5)

A fundamental theorem of B-homomorphism for B-algebras

Noetherian Jordan Banach Algebras Are Finite-Dimensional

A homomorphism theorem for partial algebras

Homomorphisms of certain Banach function algebras

A characterization of commutative Fréchet algebras with all ideals closed