Show that R can be written as a direct product of two or more (nonzero) rings iff R cont

Alaina Marshall 2022-05-24 Answered
Show that R can be written as a direct product of two or more (nonzero) rings iff R contains a non-trivial idempotent. Show that if e is an idempotent, then R = R e × R ( 1 e ) and that R e may be realized as a localization, R e = R [ e 1 ].
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Answers (1)

nicoupsqb
Answered 2022-05-25 Author has 5 answers
Assume A A 1 A 2 . It follows e := ( 1 , 0 ) 1 is a non trivial idempotent. Assume conversely that 1 e A is a non trivial idempotent. Let I := ( e ) , J := ( e 1 ) A be the ideal generated by e , e 1. It follows I + J = ( 1 ) hence these ideals are coprime. Moreover I J = I J. Since I J := A ( e ( e 1 ) ) = A ( e 2 e ) = A ( 0 ) = ( 0 ) is the zero ideal, we get by the CRT an isomorphism is the zero ideal, we get by the CRT an isomorphism
A A / I J A / I × A / J
where A / I , A / J 0. Hence by the CRT we get that A has a non-trivial idempotent iff A may be written as a (non-trivial) direct sum of two rings.
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