Domain Abstract Algebra

These are abstract structures which appear in many different branches of mathematics including geometry number theory topology and more.

Domain abstract algebra. In mathematics and more specifically in algebra a domain is a nonzero ring in which ab 0 implies a 0 or b 0. Integral domains are essentially rings without any zero divisors. In this case i m using the term properties the same way someone talking about property based.

Abstract algebra deals with groups rings fields and modules. They even appear in scientific topics such as quantum mechanics. Equivalently a domain is a ring in which 0 is the only left zero divisor.

A commutative domain is called an integral domain. Definition an integral domain is a commutative ring with such that for all the statement implies either or. An equivalent definition is as follows.

Definition given a ring r displaystyle r a zero divisor is an element a r displaystyle a in r such that x r x 0 displaystyle exists x in r x neq 0 such that a x 0 r displaystyle a x 0 r. Mathematical literature contains multiple variants of the definition of domain.