Semiring Theory

Semirings are ring-like algebraic structures where subtraction is either impossible or disallowed. They are interesting generalizations of rings and distributive lattices and have important applications in many branches of science and engineering.

A course on semiring theory may include the following topics:

  • Definition of semirings and examples

  • Building new semirings from old examples

  • Distinguished elements of semirings (zero-divisors, nilpotents, idempotents, units, complemented elements, and so on)

  • Ideal theory in semirings (prime, semiprime, primary, and maximal ideals)

  • Factor semirings

  • Morphisms of semirings and their kernels

  • Semirings of fractions (quotient theory): Bourne, Iizuka, and partitioning

  • Semimodules over semirings

  • Factor semimodules

An advanced course on semirings may include the following additional topics:

  • Linear algebra over semirings

  • Partially-ordered semirings

  • Complete semirings

  • Complete semimodules

For more on semiring theory refer to the following book:

Golan, J. S. (2013). Semirings and their Applications. Springer Science & Business Media.

Other courses on algebra:

For a brief history of algebra, see the following page: