Exercise 1170Algebra › Rings and fieldsL1Added 3 Nov 2026

Ideals

#ideal#definition#examples

Statement

Let $R$ be a commutative ring. An ideal $I\subseteq R$ is an additive subgroup such that $rx\in I$ for all $r\in R$ , $x\in I$ . Show that:

(a) $n\mathbb{Z}$ is an ideal of $\mathbb{Z}$ ;
(b) the set of polynomials with zero constant term is an ideal of $\mathbb{R}[X]$ ;
(c) $\{0\}$ and $R$ are ideals.

Hint

Check closure under addition and absorption by multiplication.

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Solution

(a) $n\mathbb{Z}$ is an additive subgroup of $\mathbb{Z}$ . For $r\in\mathbb{Z}$ and $x=nk$ , $rx=rnk=n(rk)\in n\mathbb{Z}$ . Ideal.

(b) Let $I=\{P\in\mathbb{R}[X]:P(0)=0\}$ . Sum of two such polynomials has value $0$ at $0$ ; product $QP$ has $(QP)(0)=Q(0)\cdot0=0$ . Ideal (the kernel of evaluation at $0$ ).

(c) $\{0\}$ : trivially an ideal. $R$ : trivially an ideal. These are the trivial ideals.

Conclusion. All three are ideals.

Ideals

Statement

Solution

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