Search Results for ""

The integral closure of a commutative unit ring R in an extension ring S is the set of all elements of S which are integral over R. It is a subring of S containing R.

A ring that is commutative under multiplication, has a multiplicative identity element, and has no divisors of 0. The integers form an integral domain.

Given a commutative unit ring R and an extension ring S, an element s of S is called integral over R if it is one of the roots of a monic polynomial with coefficients in R.

An extension ring R subset= S such that every element of S is integral over R.

The interior of a set is the union of all its open subsets. More informally, the interior of geometric structure is that portion of a region lying "inside" a specified ...

Given a smooth function f:R^n->R^n, if the Jacobian is invertible at 0, then there is a neighborhood U containing 0 such that f:U->f(U) is a diffeomorphism. That is, there is ...

For {M_i}_(i in I) a family of R-modules indexed by a directed set I, let sigma_(ji):M_j->M_i i<=j be an R-module homomorphism. Call (M_i,sigma_(ji)) an inverse system over I ...

An invertible linear transformation T:V->W is a map between vector spaces V and W with an inverse map which is also a linear transformation. When T is given by matrix ...

A polynomial admitting a multiplicative inverse. In the polynomial ring R[x], where R is an integral domain, the invertible polynomials are precisely the constant polynomials ...

An involutive algebra is an algebra A together with a map a|->a^* of A into A (a so-called involution), satisfying the following properties: 1. (a^*)^*=a. 2. (ab)^*=b^*a^*. ...