Semirings

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, which permits generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The pair’’ $(\text{mcA},{\text{mcA}}_0)$ is studied along the lines of universal algebra.