Baer multiplication

A binary operation on the set of classes of extensions of modules, proposed by R. Baer [1]. Let $ A $ and $ B $ be arbitrary modules. An extension of $ A $ with kernel $ B $ is an exact sequence:

$$ \tag{1 } 0 \rightarrow B \rightarrow X \rightarrow A \rightarrow 0. $$

The extension (1) is called equivalent to the extension

$$ 0 \rightarrow B \rightarrow X _ {1} \rightarrow A \rightarrow 0 $$

if there exists a homomorphism $ \alpha : X \rightarrow X _ {1} $ forming part of the commutative diagram

$$ \begin{array}{ccccc} {} &{} & X &{} &{} \\ {} &\nearrow &{} &\searrow &{} \\ B &{} &\downarrow &{} & A \\ {} &\searrow &{} &\nearrow &{} \\ {} &{} &X _ {1} &{} &{} \\ \end{array} $$

The set of equivalence classes of extensions is denoted by $ \mathop{\rm Ext} (A, B) $. The Baer multiplication on $ \mathop{\rm Ext} (A, B) $ is induced by the operation of products of extensions defined as follows. Let

$$ \tag{2 } 0 \rightarrow B \mathop \rightarrow \limits ^ \beta X \mathop \rightarrow \limits ^ \alpha A \rightarrow 0, $$

$$ \tag{3 } 0 \rightarrow B \rightarrow ^ { {\beta _ 1} } Y \rightarrow ^ { {\alpha _ 1} } A \rightarrow 0 $$

be two extensions. In the direct sum $ X \oplus Y $ the submodules

$$ C = \{ {(x, y) } : { \alpha (x) = \alpha _ {1} (y) } \} $$

and

$$ D = \{ {(-x, y) } : { x = \beta (b),\ y = \beta _ {1} (b) } \} $$

are selected. Clearly, $ D \subset C $, so that one can define the quotient module $ Z = C/D $. The Baer product of the extensions (2) and (3) is the extension

$$ 0 \rightarrow B \rightarrow ^ { {\beta _ 2} } Z \rightarrow ^ { {\alpha _ 2} } A \rightarrow 0, $$

where

$$ \beta _ {2} (b) = \ [ \beta (b), 0] = \ [0, \beta ^ \prime (b)], $$

and

$$ \alpha _ {2} [x, y] = \ \alpha (x) = \ \alpha _ {1} (y). $$

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