Two-sided ideal

A two-sided ideal

I

R I \subseteq I

and

I R \subseteq I

Square-zero ideal

I

satisfying

I^{2} = 0

, so every product of two elements of

I

vanishes.

I^{2} = 0

, then

1 + I = {1 + a : a \in I}

is an abelian subgroup of the unit group. Multiplication satisfies

(1 + a) (1 + b) = 1 + a + b

, and

(1 + a)^{- 1} = 1 - a

, so

a \mapsto 1 + a

identifies the additive group of

I

with

1 + I

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