Glossary of Ring Theory - Types of Elements

Types of Elements

Central

An element r of a ring R is central if xr = rx for all x in R. The set of all central elements forms a subring of R, known as the center of R.

Divisor

In an integral domain R, an element a is called a divisor of the element b (and we say a divides b) if there exists an element x in R with ax = b.

Idempotent

An element r of a ring is idempotent if r2 = r.

Integral element

For a commutative ring B containing a subring A, an element b is integral over A if it satisfies a monic polynomial with coefficients from A.

Irreducible

An element x of an integral domain is irreducible if it is not a unit and for any elements a and b such that x=ab, either a or b is a unit. Note that every prime element is irreducible, but not necessarily vice versa.

Prime element

An element x of an integral domain is a prime element if it is not zero and not a unit and whenever x divides a product ab, x divides a or x divides b.

Nilpotent

An element r of R is nilpotent if there exists a positive integer n such that rn = 0.

Unit or invertible element

An element r of the ring R is a unit if there exists an element r−1 such that rr−1=r−1r=1. This element r−1 is uniquely determined by r and is called the multiplicative inverse of r. The set of units forms a group under multiplication.

Zero divisor

A nonzero element r of R is a zero divisor if there exists a nonzero element s in R such that sr=0 or rs=0. Some authors opt to include zero as a zero divisor.