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Exercise 2.1.5 in Berrick and Keating: An Introduction to Rings and Modules reads as follows:
Let $$M$$ be an abelian group with $$Mc = 0$$ for some positive integer $$c$$, and put $$c = ab$$ for coprime integers $$a,b$$.
Write $$1 = ar + bs$$, and define endomorphisms $$\alpha$$ and $$\beta$$ of $$M$$ by:
$$\alpha (m) = arm $$
and
$$\beta (m) = bsm$$.
Verify that $$\{ \alpha , \beta \}$$ is a set of projections for the direct sum decomposition $$M = Ma \oplus Mb$$ of $$M$$.
Hence, find the full set of orthogonal idempotents of End$$(M)$$ corresponding to the decomposition $$M = M_1 \oplus M_2 \ ... \ ... \ \oplus M_k$$.
Can someone please help me get started on this problem?
Help would be appreciated.
Peter
Let $$M$$ be an abelian group with $$Mc = 0$$ for some positive integer $$c$$, and put $$c = ab$$ for coprime integers $$a,b$$.
Write $$1 = ar + bs$$, and define endomorphisms $$\alpha$$ and $$\beta$$ of $$M$$ by:
$$\alpha (m) = arm $$
and
$$\beta (m) = bsm$$.
Verify that $$\{ \alpha , \beta \}$$ is a set of projections for the direct sum decomposition $$M = Ma \oplus Mb$$ of $$M$$.
Hence, find the full set of orthogonal idempotents of End$$(M)$$ corresponding to the decomposition $$M = M_1 \oplus M_2 \ ... \ ... \ \oplus M_k$$.
Can someone please help me get started on this problem?
Help would be appreciated.
Peter
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