 Home

Codex Wiki

Mathematics

Area of mathematics

Algebra

Commutative algebra

A commutative ring $R$ has a nontrivial idempotent $e$ exactly when it decomposes as a product of two nontrivial rings. The isomorphism determined by $e$ is
$$
R\longrightarrow eR\times(1-e)R,
\qquad r\longmapsto(er,(1-e)r),
$$
with inverse $(x,y)\mapsto x+y$.


Ring product decomposition by an idempotent

Disconnected reduced spectrum product decomposition

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 4 / 1 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 1E / ii / Solution

Ring product decomposition by an idempotent

 Ancestors (5)

 Incoming links (3)

 Discussion (0)

 Articles by others on the same topic (0)

Ring product decomposition by an idempotent

 Ancestors (5)

 Incoming links (3)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)