Radical ideal

Definition

Equivalent definitions in tabular format

No.	Shorthand	An ideal in a commutative unital ring is termed a radical ideal if ...	An ideal $I$ in a commutative unital ring $R$ is termed a radical ideal if ...
1	closed under taking roots	whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal	For any $a\in R$ and any positive integer $n$, if $a^n\in I$, then $a\in I$.
2	equals its own radical	it equals its own radical in the whole ring	$I=\sqrt{I}$ where $\sqrt{}$ denotes the radical of an ideal, i.e., the set of all elements for which some positive power lies inside the ideal.
3	quotient ring is reduced	the quotient ring by the ideal has trivial nilradical (that is, it is a reduced ring)	the quotient ring $R/I$ is a reduced ring: whenever $x\in R/I$ and $n$ is a positive integer such that $x^n=0$, then $x=0$.
4	intersection of prime ideals	it can be expressed as an intersection of prime ideals. The intersection is allowed to be finite or infinite. An empty intersection, which would give the whole ring, is also allowed.	There exists a collection $P_s,s\in S$ of prime ideals indexed by a set $S$ such that $I={\bigcap }_{s\in S}{P}_s$. $S$ is allowed to be finite or infinite, and is also allowed to be empty.

Note that unlike for the definition of prime ideals, we do not require a radical ideal to be proper, so the whole ring is always a radical ideal in itself.

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings

This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: reduced ring | View other quotient-determined properties of ideals in commutative unital rings

Relation with other properties

Stronger properties

• Maximal ideal
• Prime ideal

Incomparable properties

• Irreducible ideal
• Primary ideal

Metaproperties

Metaproperty	Satisfied?	Proof	Statement with symbols
intersection-closed property of ideals in commutative unital rings	Yes	intersection of radical ideals is radical	Suppose $I_s,s\in S$ is a (possibly finite, possibly infinite) collection of radical ideals in a commutative unital ring $R$. Then the intersection ${\bigcap }_{s\in S}{I}_s$ is also a radical ideal in $R$.
contraction-closed property of ideals in commutative unital rings	Yes	Fill this in later	If $f:R\to S$ is a homomorphism of commutative unital rings, and $I$ is a radical ideal in $S$, then $f^{-1}(I)$ is a radical ideal in $R$.
intermediate subring condition for ideals	Yes	Fill this in later	Suppose $I$ is a radical ideal in a commutative unital ring $R$ and $S$ is a unital subring of $R$ that contains $I$. Then, $I$ is also a radical ideal in $S$.
transfer condition for ideals	Yes	Fill this in later	If $I$ is a radical ideal in $R$, and $S$ is a subring of $R$, then $I\cap S$ is a radical ideal in $S$.