Let \(R\) be a commutative ring with \(1\). The ideals of \(R\) are exactly the kernels of ring homomorphisms out of \(R\), and the isomorphism theorems hold for quotient rings. An ideal \(P\) is prime if and only if \(R/P\) is an integral domain, and maximal if and only if \(R/P\) is a field; so every maximal ideal is prime, and \((0)\subseteq\ZZ\) is prime and not maximal. By Zorn’s lemma (preliminaries), every proper ideal of \(R\) is contained in a maximal ideal.
Ring (Tag D-GURUB)
A ring is a triple \((R, +, \cdot)\) consisting of a set \(R\) and binary operations \(+\) and \(\cdot\) on \(R\) such that
\((R, +)\) is an abelian group, with identity \(0\);
\((R, \cdot)\) is a monoid, with identity \(1\);
for all \(a, b, c\in R\), \(a(b+c) = ab + ac\) and \((b+c)a = ba + ca\).
A ring is commutative if additionally \(ab = ba\) for all \(a, b\in R\).
Ring morphism (Tag D-GXMDW)
Let \(R\) and \(S\) be rings. A ring morphism is a map \(f\colon R\to S\) such that for all \(a, b\in R\):
\(f(1_R) = 1_S\);
\(f(a+b) = f(a) + f(b)\);
\(f(ab) = f(a)f(b)\).
Subring criterion (Tag PR-PZZUO)
Let \(R\) be a ring. A subset \(S\subseteq R\) is a subring of \(R\) if and only if
\((S, +)\) is a subgroup of \((R,+)\): \(0\in S\), and \(a-b\in S\) for all \(a,b\in S\); and
\((S, \cdot)\) is a submonoid of \((R,\cdot)\): \(1\in S\), and \(ab\in S\) for all \(a,b\in S\).
Units (Tag D-QQIQZ)
Let \(R\) be a ring. An element \(r\in R\) is a unit if there exists \(s\in R\) such that \(rs = sr = 1\). The set of units of \(R\) is denoted \(R^{\times}\).
If \(r\in R^{\times}\), the element \(s\) with \(rs=sr=1\) is unique: if also \(rs'=s'r=1\), then \(s'=s'(rs)=(s'r)s=s\). It is written \(r^{-1}\), and \((R^{\times}, \cdot)\) is a group, abelian when \(R\) is commutative.
Zero divisor (Tag D-4I3SL)
Let \(R\) be a ring. An element \(r\in R\) is a zero divisor if there exists \(a\in R\setminus\theset 0\) such that \(ar = ra = 0\).
If \(R\) is commutative, then \(r\in R\) is a zero divisor if and only if the \(R\)-linear map \[ R \to R, \qquad x \mapsto rx \] is not injective.
Zero divisor (Tag FD-S62UB)
Let \(R\) be a commutative ring. An element \(r\in R\) is a zero divisor if there exists a nonzero \(x\in R\) such that \(rx=0\).
Equivalently, the \(R\)-linear map \(R\to R\), \(x\mapsto rx\), is not injective, since its kernel is \(\theset{x\in R \st rx=0}\).
Integral domain (Tag D-QJ3QL)
A nonzero commutative ring \(R\) is an integral domain if it has no nonzero zero divisors: \[ \text{for all } a, b\in R,\quad ab = 0 \implies a = 0 \text{ or } b = 0. \]
Division ring (Tag D-7O2CH)
A division ring, or skew field, is a ring \(R\), not necessarily commutative, with \(1\neq 0\) in which every nonzero element is a unit, that is, \(R\setminus\theset 0 \subseteq \unitsof{R}\).
Field (Tag D-UI6CU)
A field is a commutative division ring: a commutative ring with \(1\neq 0\) in which every nonzero element is a unit.
Characteristic (Tag D-JNCUB)
Let \(R\) be a ring. The characteristic of \(R\) is the smallest integer \(n\geq 1\) such that \(\sum_{k=1}^n 1_R = 0\) if such an integer exists, and \(0\) otherwise.
Prime subfields are $\QQ$ or $\FF_p$ (Tag T-U3EZL)
Let \(F\) be a field. Its prime subfield is isomorphic to \(\QQ\) if \(\ch F=0\), and to \(\FF_p\) if \(\ch F=p>0\), in which case \(p\) is prime.
Prime subfield (Tag D-EOCCU)
Let \(F\) be a field. The prime subfield of \(F\) is the intersection of all subfields of \(F\), that is, the smallest subfield of \(F\) containing \(1\).
Ideal (Tag D-GOFWL)
Let \(R\) be a commutative ring. An ideal \(I\normal R\) is a subset \(I\subseteq R\) such that \((I, +)\) is a subgroup of \((R, +)\) and \(ri \in I\) for all \(r\in R\) and \(i\in I\).
Let \(R\) be a commutative ring. A nonempty subset \(I\subseteq R\) is an ideal if and only if \(I + I \subseteq I\) and \(RI \subseteq I\).
Ideal generated by a subset (Tag D-A7MC4)
Let \(R\) be a commutative ring and let \(S\subseteq R\) be a subset. The ideal generated by \(S\) is \[ \gens{S} \coloneqq \theset{ \sum_{i=1}^n r_i s_i \st n\geq 0,\ r_i\in R,\ s_i\in S }. \]
For \(a, b\in R\), \[ \gens{a, b} = Ra + Rb = \theset{ r_1 a + r_2 b \st r_1, r_2 \in R }. \]
Principal ideal (Tag D-D7VK2)
Let \(R\) be a commutative ring. An ideal \(I \normal R\) is principal if there exists \(a\in R\) such that \(I = \gens{a} = Ra\).
Sum, product, and intersection of ideals (Tag PR-ZDWKC)
Let \(R\) be a commutative ring and let \(I, J\subseteq R\) be ideals.
\(I + J = \theset{i+j \suchthat i\in I, j\in J}\) is the smallest ideal containing \(I\) and \(J\).
\(IJ \coloneqq \theset{\sum_{k=1}^N x_k y_k \suchthat N\geq 0,\ x_k\in I,\ y_k \in J}\) is an ideal, the ideal generated by the products \(xy\) with \(x\in I\) and \(y\in J\), and \(IJ\subseteq I\cap J\).
\(I \cap J\) is an ideal.
If \(I+J = R\), then \(I \cap J = IJ\).
(a)–(c) Each set contains \(0\) and is closed under subtraction and under multiplication by elements of \(R\); any ideal containing \(I\) and \(J\) contains all sums \(i+j\), and any ideal containing all products \(xy\) contains their finite sums. Each product \(xy\) lies in \(I\) and in \(J\), so \(IJ\subseteq I\cap J\).
- Write \(1=i+j\) with \(i\in I\) and \(j\in J\). For \(x\in I\cap J\), \(x=xi+xj\), where \(xi\in JI=IJ\) and \(xj\in IJ\), so \(x\in IJ\).
A union of ideals need not be an ideal: in \(\ZZ\), \(2\in 2\ZZ\cup3\ZZ\) and \(3\in2\ZZ\cup3\ZZ\), but \(2+3=5\notin2\ZZ\cup3\ZZ\).
Proper ideals contain no units (Tag PR-RHQZT)
Let \(R\) be a ring and \(I\subseteq R\) an ideal. Then \(I\neq R\) if and only if \(I\) contains no unit of \(R\).
If \(I=R\), then the unit \(1\) lies in \(I\). If \(u\in I\) is a unit, then \(1=u^{-1}u\in I\), so \(r=r\cdot1\in I\) for every \(r\in R\) and \(I=R\).
Prime ideal (Tag D-5BM46)
Let \(R\) be a commutative ring. A proper ideal \(\mfp \normal R\) is prime if for all \(a, b\in R\), \[ ab\in \mfp \implies a\in \mfp \text{ or } b\in \mfp. \]
Maximal ideal (Tag D-7XH2R)
Let \(R\) be a ring. A proper ideal \(\mfm \normal R\) is maximal if for every ideal \(I \normal R\) with \(\mfm \subsetneq I\), we have \(I = R\).
Maximal, prime, and radical ideals via their quotient rings (Tag PR-76BDN)
Let \(R\) be a commutative ring, and let \(\mfm\), \(\mfp\), and \(J\) be ideals of \(R\).
\(R/\mfm\) is a field if and only if \(\mfm\) is a maximal ideal.
\(R/\mfp\) is an integral domain if and only if \(\mfp\) is a prime ideal.
Prime spectrum (Tag D-CXXCG)
Let \(R\) be a commutative ring. The prime spectrum, or spectrum, of \(R\) is the set of prime ideals of \(R\): \[ \Spec(R) \coloneqq \theset{\mfp \normal R \suchthat \mfp \text{ is prime}}. \]
Maximal spectrum (Tag D-NX4KW)
Let \(R\) be a commutative ring. The max spectrum of \(R\) is the set of maximal ideals of \(R\): \[ \mspec(R) \coloneqq \theset{\mfm \subseteq R \suchthat \mfm \text{ is a maximal ideal}}. \]
First isomorphism theorem for rings (Tag PR-Z3YWJ)
Let \(f\colon A\to B\) be a ring homomorphism. Then \(\ker f\) is an ideal of \(A\), \(\im f\) is a subring of \(B\), and \(f\) induces a ring isomorphism \[ \bar f\colon A/\ker f \to \im f,\qquad a+\ker f\mapsto f(a). \] In particular, if \(f\) is surjective, then \(A/\ker f \cong B\). As abelian groups, \(0 \to \ker f \to A \mapsvia{f} \im f \to 0\) is exact.
If \(x,y\in\ker f\) and \(a\in A\), then \(f(x-y)=0\) and \(f(ax)=f(a)f(x)=0=f(x)f(a)=f(xa)\), so \(\ker f\) is an ideal. The image contains \(1_B=f(1_A)\) and is closed under subtraction and multiplication, so it is a subring. The map \(\bar f\) is well defined and injective because \(f(a)=f(a')\) if and only if \(a-a'\in\ker f\); it is surjective onto \(\im f\) by construction, and it is a ring homomorphism because \(f\) is.
Second isomorphism theorem for rings (Tag PR-GGCEU)
Let \(R\) be a ring, \(S \subseteq R\) a subring, and \(I \subseteq R\) a two-sided ideal. Then \(S + I\) is a subring of \(R\), \(S \intersect I\) is a two-sided ideal of \(S\), and there is a ring isomorphism \[ \frac{S+I}{I} \cong \frac{S}{S \intersect I}. \]
Correspondence theorem for ideals (Tag PR-LJE4C)
Let \(R\) be a ring, \(I \subseteq R\) a two-sided ideal, and \(\phi\colon R \to R/I\) the quotient map.
The maps \(J \mapsto \bar{J} \coloneqq \phi(J) = J/I\) and \(\bar{J} \mapsto \inverseof{\phi}(\bar{J})\) are mutually inverse inclusion-preserving bijections \[ \begin{aligned} \correspond{\text{ideals } J \subseteq R \text{ with } J \contains I} &\mapstofrom \correspond{\text{ideals of } R/I} \\ J &\mapsto J/I \\ \inverseof{\phi}(\bar J) &\mapsfrom \bar{J}. \end{aligned} \] In particular, every ideal of \(R/I\) is of the form \(S/I\) for a unique ideal \(S \subseteq R\) containing \(I\).
The same maps give a bijection between subrings \(A \subseteq R\) containing \(I\) and subrings of \(R/I\).
(Third isomorphism theorem.) For ideals \(I \subseteq J \subseteq R\), there is a ring isomorphism \[ \frac{R/I}{J/I} \cong \frac{R}{J}. \]
A commutative ring is a field if and only if its only ideals are $0$ and $R$ (Tag PR-DDDXH)
Let \(R\) be a commutative ring with \(1\). Then \(R\) is a field if and only if the set of ideals of \(R\) is exactly \(\theset{0, R}\) with \(0 \neq R\).
Simple ring (Tag FD-SNOTW)
A nonzero ring \(R\) is simple if its only two-sided ideals are \((0)\) and \(R\).
Wedderburn's little theorem (Tag T-GYVNQ)
Every finite division ring is commutative, and hence a field.