Associative rings and algebras Definition

Nil ideal

Builds on Ring Nilpotent element Left ideal Two-sided ideal

Definition

Let $R$ be a ring. Let $I$ be a left ideal, right ideal or two-sided ideal of $R$ . The ideal $I$ is nil if every element of $I$ is nilpotent. This means that for each $a \in I$ there is an integer $n \ge 1$ with $a^n = 0$ .

Remarks

Example: in $\Z/8\Z$ , the ideal $I = 2\Z/8\Z$ is nil. The check: every element of $I$ is $2b$ for some residue $b$ , and $(2b)^3 = 8b^3 = 0$ . Non-example: in $\Z$ , the ideal $2\Z$ is not nil, since $2^n \ne 0$ for all $n \ge 1$ . In a commutative ring the nil ideals are the ideals contained in the nilradical. The exponent $n$ may depend on the element, and there need not be a common bound.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Group concept
Left ideal concept
Ring definition
Ideal definition
Nilpotent element definition
Two-sided ideal definition
Nil ideal definition

Tag	`WKG2DZDEBT`
URI	`m2://WKG2DZDEBT`
Kind	definition
Writing	W0 stub
MSC	16N40
Defines	Nil ideal
Uses	Ring, Nilpotent element, Left ideal, Two-sided ideal
Changed	2 November 2026

Nil ideal

Definition

Remarks

Reading path

Further reading

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