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For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x. If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring. If a ring R contains the zero ring as a subring, then R itself is the zero ring. [6]
If R and S are rings and there exists a ring homomorphism R → S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the zero ring, which has only a single element 0.
The unit group of the ring M n (R) of n × n matrices over a ring R is the group GL n (R) of invertible matrices. For a commutative ring R, an element A of M n (R) is invertible if and only if the determinant of A is invertible in R. In that case, A −1 can be given explicitly in terms of the adjugate matrix.
The unit group of M n (R), consisting of the invertible matrices under multiplication, is denoted GL n (R). If F is a field, then for any two matrices A and B in M n (F), the equality AB = I n implies BA = I n. This is not true for every ring R though. A ring R whose matrix rings all have the mentioned property is known as a stably finite ring ...
Formally, the polynomial ring in n noncommuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst ...
The zero map R → S that sends every element of R to 0 is a ring homomorphism only if S is the zero ring (the ring whose only element is zero). For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object in the category of rings. For every ring R, there is a unique ring homomorphism ...
1. A ring R is commutative if the multiplication is commutative, i.e. rs = sr for all r,s ∈ R. 2. A ring R is skew-commutative ring if xy = (−1) ε(x)ε(y) yx, where ε(x) denotes the parity of an element x. 3. A commutative algebra is an associative algebra that is a commutative ring. 4. Commutative algebra is the theory of commutative rings.
If R is a regular domain (i.e., regular at any prime ideal), then Pic(R) is precisely the divisor class group of R. [7] For example, if R is a principal ideal domain, then Pic(R) vanishes. In algebraic number theory, R will be taken to be the ring of integers, which is Dedekind and thus regular.