Topological ring . It is a set provided with two mathematical structures. In addition to topological groups, in mathematics , topological rings and bodies are defined, that is, algebraic rings and bodies with continuous operations.
A set A is called a topological ring if
- A is a ring
- A is a topological space
- The existing algebraic operations in A are continuous in the topological space A.
In the language of the environments for any elements a and b of A and for any environments H and K of elements ab and ab there are environments L and M of elements a and b such that LM contained in H and LM is part of K.
Summary
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- 1 Application
- 1 Homomorphism nucleus
- 2 Other morphism
- 2 Examples
- 3 Source
Application
An application h of a topological ring R on a topological ring S is called homomorphic if it is a homomorphic application of the algebraic ring R on the algebraic ring S and is a continuous application of the topological ring R on the topological ring S.
Homomorphism nucleus
The set of all the elements of the R ring that are applied by the homomorphism h at the zero of the S ring is called the nucleus of the h homomorphism. By the way, this nucleus of homomorphism is an ideal of the algebraic ring R and a closed of the topological space R.
Another morphism
An application h of a topological ring R in a topological ring S is called isomorphic if it is an isomorphic application of the algebraic ring R in the algebraic ring S and is a homeomorphism (bijective and bicontinuous) of the topological ring R in the topological ring S.
