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.
Read more about this topic: Glossary Of Ring Theory
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