I need help to resolve an apparent contradiction between part of a Proposition proved by Paul Bland in his book "Rings and Their Modules" and an Example provided by Joseph Rotman in his book "An Introduction to Homological Algebra" (Second Edition).(adsbygoogle = window.adsbygoogle || []).push({});

One element of Bland's Proposition 3.2.7 is the assertion (and proof) that

##M \cong M_1 \oplus M_2## ... ...

##\Longrightarrow##

... the short exact sequence

##0 \to M_1 \stackrel{f}{\to} M \stackrel{g}{\to} M_2 \to 0##

is split

However ... ...

Rotman in Example 2.29 (page 54) constructs a sequence

##0 \to A \stackrel{i'}{\to} A \oplus M \stackrel{p'}{\to} M \to 0##

which is not split ... ...

Thus Rotman appears to construct a counterexample to Bland's Theorem ...

BUT ...

how can this be ... ... ???

Can someone please resolve this issue ... ?

Help will be very much appreciated ... ...

Peter

Bland's Proposition 3.2.7 reads as follows:

Rotman's Example 2.29 reads as follows:

To give readers the necessary Definitions and Propositions on exact sequences in Bland I am providing the following relevant text from Bland ... ...

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# I Split Exact Sequences ... Bland, Proposition 3.2.7

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