Search results
Results from the WOW.Com Content Network
A GCD domain is integrally closed, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain. For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that dm = xy, or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then ...
In a GCD domain (for instance in ), the operations of GCD and LCM are idempotent. In a Boolean ring, multiplication is idempotent. In a Tropical semiring, addition is idempotent. In a ring of quadratic matrices, the determinant of an idempotent matrix is either 0 or 1.
A is a GCD domain satisfying ACCP. A is a Schreier domain, [6] and atomic. A is a pre-Schreier domain and atomic. A has a divisor theory in which every divisor is principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every prime ideal of height 1 ...
• GCD domain • Unique factorization domain • Principal ideal domain ... One example of an idempotent element is a projection in linear algebra.
• GCD domain • Unique factorization domain ... is an additively idempotent semifield with the semifield sum defined to be the supremum of two elements.
Idempotent analysis (1 C, 3 P) Integers (24 C, 450 P) M. Module theory (84 P) T. Theorems in ring theory (32 P) ... GCD domain; Gelfand ring; Generalized Clifford ...
An idempotent a + I in the quotient ring R / I is said to lift modulo I if there is an idempotent b in R such that b + I = a + I. An idempotent a of R is called a full idempotent if RaR = R. A separability idempotent; see Separable algebra. Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b ...
The idempotent of is a codeword such that = (that is, is an idempotent element of ) and is an identity for the code, that is = for every codeword . If n {\displaystyle n} and q {\displaystyle q} are coprime such a word always exists and is unique; [ 2 ] it is a generator of the code.