Principal ideal

In the context of abstract algebra, specifically in ring theory, a principal ideal is a specific type of ideal in a ring that can be generated by a single element. Formally, let \( R \) be a ring and let \( a \) be an element of \( R \).

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Codex  0 Created 2026-09-24 Updated 2026-11-05

(a) = R a = {r a : r \in R} .

For example, every ideal of

Z

has the form

(n)

, whereas

(X, Y)

in a two-variable polynomial ring over a field is not principal.

a

consists of finite sums

\sum_{i} r_{i} a s_{i}

; it need not equal the principal left ideal

R a

. In

M_{2} (k)

E_{11}

generates the entire matrix ring as a two-sided ideal, since

E_{i 1} E_{11} E_{1 j} = E_{ij}

. The left ideal

M_{2} (k) E_{11}

only contains matrices with zero second column. The side is therefore part of a principal-ideal statement outside the commutative setting.