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
Famous quotes containing the words types of, types and/or elements:
“The wider the range of possibilities we offer children, the more intense will be their motivations and the richer their experiences. We must widen the range of topics and goals, the types of situations we offer and their degree of structure, the kinds and combinations of resources and materials, and the possible interactions with things, peers, and adults.”
—Loris Malaguzzi (1920–1994)
“As for types like my own, obscurely motivated by the conviction that our existence was worthless if we didn’t make a turning point of it, we were assigned to the humanities, to poetry, philosophy, painting—the nursery games of humankind, which had to be left behind when the age of science began. The humanities would be called upon to choose a wallpaper for the crypt, as the end drew near.”
—Saul Bellow (b. 1915)
“Barbarisation may be defined as a cultural process whereby an attained condition of high value is gradually overrun and superseded by elements of lower quality.”
—Johan Huizinga (1872–1945)