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: