This article is about mathematics. Practically all noetherian rings that appear in application are catenary. However, commutative theory of structures lecture notes pdf can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism.

Morita equivalence is especially important in algebraic topology and functional analysis. Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed. Noncommutative rings serve as an active area of research due to their ubiquity in mathematics. Those rings are essentially the same things as varieties: they correspond in essentially a unique way.

Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century. Mathematical Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. London Mathematical Society Student Texts, 16. Cambridge University Press, Cambridge, 1989. Reprint of the 1968 original.

With an afterword by Lance W. Mathematical Association of America, Washington, DC, 1994. American Mathematical Society Colloquium Publications, Vol. Revised edition American Mathematical Society, Providence, R.

American Mathematical Society Mathematical Surveys, vol. American Mathematical Society, New York, 1943. An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Graduate Texts in Mathematics, 131.