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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules". An extension is said to be trivial or to split if ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism [ 2 ...
Actually, Carathéodory's extension theorem can be slightly generalized by replacing ring by semi-field. [2] The definition of semi-ring may seem a bit convoluted, but the following example shows why it is useful (moreover it allows us to give an explicit representation of the smallest ring containing some semi-ring).
Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E. If we used a ring, say, A as a base ring instead of Z , {\displaystyle \mathbb {Z} ,} then the resulting ring will be over A .
A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N-2 rings ...
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras.The theorem states that an (Artinian) [a] semisimple ring R is isomorphic to a product of finitely many n i-by-n i matrix rings over division rings D i, for some integers n i, both of which are uniquely determined up to permutation of the index i.
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The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b, then c n = a n + b n for every n in N.