Baer multiplication

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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:


The extension (1) is called equivalent to the extension

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


be two extensions. In the direct sum the submodules


are selected. Clearly, , so that one can define the quotient module . The Baer product of the extensions (2) and (3) is the extension




[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:
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article