Ideal (ring theory) · aiwiki.page

An ideal is a subset of a ring that is closed under addition and additive inverses and absorbs multiplication by arbitrary elements of the ring. In a commutative ring, this absorption condition means that $ra$ belongs to the ideal whenever $a$ does and $r$ belongs to the ring. In a noncommutative ring, left, right, and two-sided ideals must be distinguished. Ideals provide the subsets by which rings can be quotiented, and they generalize aspects of divisibility from integers to more general algebraic structures. (math.ucla.edu)

Definition and conventions

Let $R$ be an associative ring with identity. A two-sided ideal $I\subseteq R$ satisfies:

$I$ is an additive subgroup of $R$ .
For every $a\in I$ and $r\in R$ , both $ra\in I$ and $ar\in I$ .

Equivalently, $I$ is nonempty, is closed under subtraction, and satisfies these multiplication conditions. A left ideal requires only $ra\in I$ ; a right ideal requires only $ar\in I$ . These notions coincide when multiplication in $R$ is commutative. Unless stated otherwise, the discussion below concerns commutative rings with identity. (math.ucla.edu)

Every ring has the zero ideal $(0)=\{0\}$ and the unit ideal $R=(1)$ . An ideal is proper if it is not the whole ring. An ideal containing $1$ , or any invertible element, equals $R$ : if $u\in I$ is invertible, then $u^{-1}u=1\in I$ , and consequently every $r=r1$ lies in $I$ . Thus a proper ideal does not contain the ring’s identity, even though it is closed under its own addition and multiplication. (math.ucla.edu)

Generated ideals and examples

The ideal generated by a subset $S\subseteq R$ , denoted $(S)$ , is the smallest ideal containing $S$ . In a commutative ring it consists of all finite sums

r_1s_1+\cdots+r_ms_m, \qquad r_i\in R,\quad s_i\in S.

For finitely many generators, the notation is $(a_1,\ldots,a_m)$ . An ideal generated by one element is a principal ideal:

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

Generators need not be unique. (kconrad.math.uconn.edu)

Representative examples include:

In the ring of integers $\mathbb Z$ , every ideal is $n\mathbb Z=(n)$ for a unique nonnegative integer $n$ .
A field has only the ideals $(0)$ and the whole field, since every nonzero element is invertible.
In the polynomial ring $k[x]$ over a field, every ideal is principal.
In $k[x,y]$ , the ideal $(x,y)$ consists of polynomials with zero constant term and is not principal. Thus multiple generators can be essential. (kconrad.math.uconn.edu)

The ambient ring matters. For example, $\mathbb Z$ is an additive subgroup and a subring of $\mathbb Q$ , but it is not an ideal of $\mathbb Q$ : multiplication of $1\in\mathbb Z$ by $1/2\in\mathbb Q$ leaves $\mathbb Z$ . Absorption is stronger than closure under multiplication within the subset. (math.ucla.edu)

Quotient rings and homomorphisms

An ideal $I$ defines an equivalence relation

a\equiv b\pmod I \quad\Longleftrightarrow\quad a-b\in I.

The quotient ring $R/I$ consists of the additive cosets $a+I$ , with operations

(a+I)+(b+I)=(a+b)+I,

(a+I)(b+I)=ab+I.

The absorption property ensures that multiplication is independent of the chosen representatives. The natural projection $R\to R/I$ is a ring homomorphism whose kernel is $I$ . Conversely, the kernel of every ring homomorphism is a two-sided ideal. (kconrad.math.uconn.edu)

The first isomorphism theorem gives

R/\ker\varphi\cong\operatorname{im}\varphi

for a ring homomorphism $\varphi$ . Moreover, ideals of $R/I$ correspond bijectively to ideals of $R$ containing $I$ , through $J\mapsto J/I$ . Quotienting can therefore be interpreted as imposing algebraic relations. For instance,

\mathbb Z/(n)

is the ring used in modular arithmetic, while

\mathbb R[x]/(x^2+1)\cong\mathbb C

imposes the relation $x^2=-1$ . (math.mit.edu)

Operations on ideals

For ideals $I,J\subseteq R$ , their sum and product are

I+J=\{a+b:a\in I,\ b\in J\},

IJ= \left\{ \sum_{\nu=1}^{m}a_\nu b_\nu: a_\nu\in I,\ b_\nu\in J \right\}.

Both are ideals, as is $I\cap J$ . The product must include finite sums: the set of individual products alone need not be closed under addition. Always,

IJ\subseteq I\cap J.

Repeated multiplication defines the powers $I^n$ . Arbitrary intersections of ideals are ideals, whereas arbitrary unions generally are not. (math.mit.edu)

The ideal quotient, or colon ideal, is

(I:J)=\{r\in R:rJ\subseteq I\}.

It measures which multipliers send $J$ into $I$ ; it is an ideal of $R$ , not a quotient ring. (math.ucla.edu)

Two ideals are comaximal if $I+J=R$ . In this case $IJ=I\cap J$ . More generally, the Chinese remainder theorem states that for finitely many pairwise comaximal ideals,

R/(I_1\cdots I_m) \cong \prod_{\nu=1}^{m}R/I_\nu.

For integer ideals, sums and intersections encode familiar arithmetic:

(a)+(b)=(\gcd(a,b)),\qquad (a)\cap(b)=(\operatorname{lcm}(a,b)).

This illustrates how ideal operations generalize divisibility. (stacks.math.columbia.edu)

Prime, maximal, radical, and primary ideals

These classes describe different properties of quotient rings and factorization.

Prime and maximal ideals

A proper ideal $\mathfrak p$ is a prime ideal if

ab\in\mathfrak p \quad\Longrightarrow\quad a\in\mathfrak p\ \text{or}\ b\in\mathfrak p.

Equivalently, $R/\mathfrak p$ is an integral domain: a nonzero commutative ring without nonzero zero divisors. (stacks.math.columbia.edu)

A proper ideal $\mathfrak m$ is a maximal ideal if no proper ideal strictly contains it. Equivalently, $R/\mathfrak m$ is a field. Every maximal ideal is prime, but the converse fails. In $k[x,y]$ , for example, $(x)$ is prime because its quotient is $k[y]$ , but it is not maximal because $k[y]$ is not a field. In $\mathbb Z$ , the nonzero prime ideals are precisely $(p)$ for prime numbers $p$ , and they are maximal. (stacks.math.columbia.edu)

Every proper ideal in a commutative ring with identity is contained in a maximal ideal. The usual general existence proof uses Zorn’s lemma. (stacks.math.columbia.edu)

Radical ideals

The radical of $I$ is

\sqrt I=\{r\in R:r^n\in I\text{ for some }n\geq1\}.

An ideal is a radical ideal if $I=\sqrt I$ . Equivalently, $R/I$ has no nonzero nilpotent elements—elements whose positive powers vanish. Every prime ideal is radical. More generally,

\sqrt I= \bigcap_{\substack{\mathfrak p\supseteq I\\ \mathfrak p\text{ prime}}}\mathfrak p.

The radical $\sqrt{(0)}$ is called the nilradical of the ring. (kconrad.math.uconn.edu)

Primary ideals

A proper ideal $Q$ is a primary ideal if

ab\in Q,\quad a\notin Q \quad\Longrightarrow\quad b^n\in Q\text{ for some }n\geq1.

Equivalently, every zero divisor in $R/Q$ is nilpotent. Prime ideals are primary, and the radical of a primary ideal is prime. For example, $(p^e)\subseteq\mathbb Z$ is primary for a prime $p$ and $e\geq1$ , but is prime only when $e=1$ . (kconrad.math.uconn.edu)

Finite generation and decomposition

A commutative ring is Noetherian if every ideal is finitely generated. Equivalently, every ascending chain

I_1\subseteq I_2\subseteq I_3\subseteq\cdots

eventually stabilizes. Hilbert’s basis theorem states that a polynomial ring in finitely many variables over a Noetherian ring is again Noetherian. Consequently, ideals in $k[x_1,\ldots,x_n]$ have finite generating sets. (math.mit.edu)

The Lasker–Noether theorem states that every proper ideal in a Noetherian ring admits a finite primary decomposition:

I=Q_1\cap\cdots\cap Q_t.

This is an intersection decomposition into primary ideals, not generally a product factorization into prime ideals. Its hypotheses matter: finite generation and finite primary decomposition are not automatic in arbitrary rings. (kconrad.math.uconn.edu)

Algebraic geometry and computation

In algebraic geometry, polynomial ideals encode systems of equations. For $I\subseteq k[x_1,\ldots,x_n]$ , define

V(I)=\{a\in k^n:f(a)=0\text{ for every }f\in I\}.

Replacing equations by the ideal they generate does not change their common zeros. If $k$ is algebraically closed, Hilbert’s Nullstellensatz states

I(V(I))=\sqrt I,

where $I(V(I))$ denotes all polynomials vanishing on that zero set. Thus geometric zero sets correspond to radical ideals, rather than to arbitrary ideals. (kconrad.math.uconn.edu)

More generally, the spectrum $\operatorname{Spec}R$ is the set of prime ideals of $R$ , equipped with the Zariski topology. Its closed sets are

V(I)=\{\mathfrak p:I\subseteq\mathfrak p\}.

Here too, $V(I)=V(\sqrt I)$ . (stacks.math.columbia.edu)

For polynomial ideals over a field, a Gröbner basis provides a method for testing ideal membership: a polynomial belongs to the ideal exactly when its remainder on division by the Gröbner basis is zero. Such bases also support elimination and calculations in quotient rings. An arbitrary generating set does not generally have the same remainder criterion. (ocw.mit.edu)

Number theory and historical development

Ideal theory arose from nineteenth-century work on factorization in algebraic number rings. Richard Dedekind first published his ideal theory in 1871, developing a framework in which subsets of a ring could replace elements as the objects of factorization. (arxiv.org)

The central arithmetic result concerns Dedekind domains, including rings of algebraic integers in number fields: every nonzero proper ideal factors uniquely, up to order, as a product of nonzero prime ideals. This remains true even when elements do not have unique factorization. (jmilne.org)

Nonzero fractional ideals of a Dedekind domain form a group under multiplication. Quotienting this group by the principal fractional ideals gives the ideal class group. It measures the failure of ideals to be principal; a Dedekind domain has trivial ideal class group exactly when every ideal is principal. These factorization and invertibility properties are special to the relevant class of rings, not properties of ideals in general. (jmilne.org)

Noncommutative distinctions

In noncommutative algebra, left ideals are submodules of the left regular module ${}_RR$ , and right ideals are submodules of $R_R$ . Quotienting by a left ideal gives a left module, but the usual quotient multiplication defines a ring only for a two-sided ideal. (math.mit.edu)

For example, in the matrix ring $M_n(k)$ , with $n\geq2$ , the matrices whose first column is zero form a left ideal but not a right ideal. Nevertheless, $M_n(k)$ has only two two-sided ideals, $(0)$ and the whole ring: it is a simple ring. Thus a noncommutative ring may have many one-sided ideals while having no nontrivial two-sided ideals, and commutative characterizations of maximal ideals by field quotients do not transfer unchanged. (math.mit.edu)

References

Abstract Algebra — Chapter 3 Ring theory, Romyar Sharifimath.ucla.edu
Notes on Ideals, Keith Conradkconrad.math.uconn.edu
Algebraic Geometry, MIT lecture notesmath.mit.edu
Rings, Ideals, and Modules, Pavel Etingofmath.mit.edu
Basic notions — The Stacks Projectstacks.math.columbia.edu
The spectrum of a ring — The Stacks Projectstacks.math.columbia.edu
Chinese remainder — The Stacks Projectstacks.math.columbia.edu
Maximal Ideals in Polynomial Rings, Keith Conradkconrad.math.uconn.edu
Noetherian Rings, Keith Conradkconrad.math.uconn.edu
Lecture 14: Monomial Orderings, MIT 6.972ocw.mit.edu
Dedekind on Higher Congruences and Index Divisors, 1871 and 1878arxiv.org
Algebraic Number Theory, J. S. Milnejmilne.org