Separable completion of a ring
From Encyclopedia of Mathematics
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The completion of the topological ring 
, where 
is a topological ring and 
is the closure in 
of the zero ideal 
. The separable completion of a ring is also a topological ring and is usually denoted by 
. Every continuous homomorphism from 
into a complete separable ring 
can be uniquely extended to a continuous homomorphism 
.
In the most important case where the topology of the ring 
is linear and is defined by a fundamental system of ideals 
, the separable completion 
is canonically identified with the projective limit 
of the discrete rings 
. The separable completion of a module is achieved in the same way.
Comments
References
| [a1] | N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies |
How to Cite This Entry:
Separable completion of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_completion_of_a_ring&oldid=12455
Separable completion of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_completion_of_a_ring&oldid=12455
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article