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

Question

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 \times R (1 - e)

and that

R e

may be realized as a localization,

R e = R [e^{- 1}]

.

Answer 1

Assume

A ≅ A_{1} \oplus A_{2}

. It follows

e := (1, 0) \neq 1

is a non trivial idempotent. Assume conversely that

1 \neq e \in A

is a non trivial idempotent. Let

I := (e), J := (e - 1) \subseteq A

be the ideal generated by

e, e - 1

. It follows

I + J = (1)

hence these ideals are coprime. Moreover

I J = I \cap 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 \times A / J

where

A / I, A / J \neq 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.

Answers (1)