Baer multiplication
From Encyclopedia of Mathematics
A binary operation on the set of classes of extensions of modules, proposed by R. Baer [1]. Let and
be arbitrary modules. An extension of
with kernel
is an exact sequence:
![]() | (1) |
The extension (1) is called equivalent to the extension
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if there exists a homomorphism forming part of the commutative diagram
![]() |
The set of equivalence classes of extensions is denoted by . The Baer multiplication on
is induced by the operation of products of extensions defined as follows. Let
![]() | (2) |
![]() | (3) |
be two extensions. In the direct sum the submodules
![]() |
and
![]() |
are selected. Clearly, , so that one can define the quotient module
. The Baer product of the extensions (2) and (3) is the extension
![]() |
where
![]() |
and
![]() |
References
[1] | R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" Math. Z. , 38 (1934) pp. 374–416 |
[2] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |
How to Cite This Entry:
Baer multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baer_multiplication&oldid=14712
Baer multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baer_multiplication&oldid=14712
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article