# 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

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. V.E. Govorov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Baer_multiplication&oldid=14712

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098