Rings

Definition

1.A ring is an algebraic structure that extends the notion of a group by introducing a second binary operation. The concept arises from the observation that several fundamental objects, such as the integers, the polynomials with real coefficients, and the square matrices of a given size, share a common pattern. They admit both an addition and a multiplication, the two operations interact in a predictable way, and yet multiplication need not be commutative and need not admit inverses. A ring is a set $R$ together with two binary operations $+$ and $\cdot$ (addition and multiplication), satisfying the following axioms:

3.A ring $(R, +, \cdot)$ is called commutative when $a \cdot b = b \cdot a$ for all $a, b \in R.$ It is called a ring with unity, or unital ring, when there exists a multiplicative identity element $1 \in R$ such that $1 \cdot a = a \cdot 1 = a$ for all $a \in R.$

4.The multiplicative identity is not part of the basic ring axioms adopted here. This convention includes rings such as $2\mathbb{Z},$ which has no multiplicative identity. The natural numbers $\mathbb{N},$ on the other hand, do not form a ring under their usual operations because a positive integer has no additive inverse in $\mathbb{N}.$

Properties

5.Several consequences follow directly from the axioms. For any $a \in R,$ multiplication by the additive identity satisfies $a \cdot 0 = 0 \cdot a = 0.$ This is a consequence of distributivity:

$$a \cdot 0 = a \cdot (0 + 0) = a \cdot 0 + a \cdot 0$$

6.and the cancellation of $a \cdot 0$ from both sides through the group structure of $(R, +).$ Similarly, for all $a, b \in R$ the following identity holds:

$$(-a) \cdot b = a \cdot (-b) = -(a \cdot b)$$

7.In particular, $(-1) \cdot a = -a$ whenever $R$ has a unity. These sign rules hold in any ring.

8.Suppose that $R$ has a unity. An element $u \in R$ is called a unit, or an invertible element, when some $v \in R$ satisfies $u \cdot v = v \cdot u = 1.$ Associativity implies that such an element $v$ is unique, so it is denoted by $u^{-1}.$ The units form a group under multiplication, usually denoted by $R^\times.$ The precise description of this group depends on the ring:

10.For the last assertion, Bezout's identity gives integers $r$ and $s$ such that $ra + sn = 1$ whenever $\gcd(a,n) = 1.$ Reduction modulo $n$ then gives $[r][a] = [1].$ Conversely, if $[r][a] = [1],$ every common divisor of $a$ and $n$ divides $1,$ so $a$ and $n$ are relatively prime.

11.A nonzero element $a \in R$ is called a zero divisor when there exists a nonzero element $b \in R$ such that $a \cdot b = 0$ or $b \cdot a = 0.$ Zero divisors are a feature that distinguishes rings from fields: in a field, no nonzero element can be a zero divisor. A commutative ring with unity that contains no zero divisors is called an integral domain.

Algebraic hierarchy

12.A group is a set equipped with a single binary operation that satisfies closure, associativity, the existence of an identity element, and the existence of inverses.

13.A ring extends this framework by introducing a second operation, multiplication, required to be associative and to distribute over addition, but not necessarily commutative and not required to admit inverses.

14.More precisely, a field is a commutative ring with unity such that $1 \neq 0$ and every nonzero element has a multiplicative inverse. The condition $1 \neq 0$ excludes the zero ring, in which the additive and multiplicative identities coincide. The three structures form a chain of increasing rigidity:

16.

The integers $\mathbb{Z}$ are the most natural example of a ring that is not a field: every integer has an additive inverse, yet $2^{-1}$ does not belong to $\mathbb{Z}.$ The rational numbers $\mathbb{Q},$ by contrast, form a field. The same hierarchy reappears in linear algebra: scalars drawn from a ring give rise to a module, and scalars drawn from a field give rise to a vector space.

Examples

17.The set $\mathbb{Z}$ of integers, equipped with ordinary addition and multiplication, is the simplest example of a commutative ring with unity. The additive identity is $0,$ the multiplicative identity is $1,$ and every integer has an additive inverse. The integers form an integral domain, since the product of two nonzero integers is always nonzero.

18.The familiar number systems $\mathbb{Q},$ $\mathbb{R},$ and $\mathbb{C}$ are commutative rings with unity. Every nonzero element in each of them has a multiplicative inverse, so they are fields.

19.The set of polynomials with real coefficients, denoted $\mathbb{R}[x],$ forms a commutative ring with unity under the usual addition and multiplication of polynomials. The additive identity is the zero polynomial, and the multiplicative identity is the constant polynomial $1.$ This ring is also an integral domain. The same construction works for polynomials in several variables. For example, $F[x,y]$ is a commutative ring with unity for every field $F.$

20.Let $X$ be a set. The real-valued functions on $X$ form a commutative ring when addition and multiplication are defined pointwise:

$$(f + g)(x) = f(x) + g(x)$$

$$(fg)(x) = f(x)g(x)$$

21.The zero function is the additive identity, and the constant function with value $1$ is the multiplicative identity. A function $f$ is a unit precisely when $f(x) \neq 0$ for every $x \in X,$ because its inverse must be the function $x \mapsto 1/f(x).$

22.Let $n$ be a positive integer. The quotient $\mathbb{Z}/n\mathbb{Z}$ consists of the residue classes $[0], [1], \ldots, [n - 1].$ Equipped with addition and multiplication modulo $n,$ it forms a commutative ring with unity. For example, in $\mathbb{Z}/6\mathbb{Z}$ one has $[2][3] = [0],$ so $[2]$ and $[3]$ are zero divisors and $\mathbb{Z}/6\mathbb{Z}$ is not an integral domain. When $n$ is prime, however, $\mathbb{Z}/n\mathbb{Z}$ contains no zero divisors and is in fact a field. The arithmetic of residue classes, together with the addition and multiplication tables that realise these operations concretely, is treated on the page about the modulo operator.

23.Let $F$ be a field and let $n$ be a positive integer. The set $\mathrm{M}_n(F)$ of all $n \times n$ matrices with entries in $F$ forms a ring under matrix addition and multiplication. The additive identity is the zero matrix, and the multiplicative identity is the identity matrix $I_n.$ For $n \geq 2,$ this ring is not commutative, since matrix multiplication does not commute in general, and it contains zero divisors. Replacing $F$ by $\mathbb{Z}$ gives the ring $\mathrm{M}_n(\mathbb{Z})$ of matrices with integer entries, so the coefficient set need not be a field.

Subrings

24.A subset $S$ of a ring $R$ is called a subring when $S$ is itself a ring under the operations inherited from $R.$ A nonempty subset $S \subseteq R$ is a subring of $R$ if and only if it is closed under subtraction and under multiplication, that is, for all $a, b \in S$ one has $a - b \in S$ and $a \cdot b \in S.$ Closure under subtraction is equivalent to requiring that $S$ be a subgroup of $(R, +),$ and closure under multiplication then ensures that the second operation is also well-defined on $S.$ Every ring $R$ contains the following canonical subrings:

26.Any subring other than $R$ is called a proper subring.

27.These two subrings coincide when $R$ is the zero ring. As an example of a proper subring, the set of even integers $2\mathbb{Z} = \{\ \ldots, -4, -2, 0, 2, 4, \ldots \ \}$ is a subring of $(\mathbb{Z}, +, \cdot).$ For any two even integers $a = 2m$ and $b = 2k,$ one has $a - b = 2(m - k) \in 2\mathbb{Z}$ and $a \cdot b = 4mk \in 2\mathbb{Z},$ so both conditions are satisfied. The set $2\mathbb{Z}$ does not contain the multiplicative identity $1$ of $\mathbb{Z},$ which illustrates that a subring of a unital ring need not itself be unital.

28.Subrings also arise by restricting entries or by imposing conditions that are preserved by the ring operations. The inclusion $\mathbb{Q} \subseteq \mathbb{R}$ makes $\mathrm{M}_n(\mathbb{Q})$ a subring of $\mathrm{M}_n(\mathbb{R}).$ The upper triangular matrices in $\mathrm{M}_n(\mathbb{R})$ form another subring because sums, additive inverses, and products of upper triangular matrices remain upper triangular. Similarly, the continuous functions from $\mathbb{R}$ to $\mathbb{R}$ form a subring of the ring of all real-valued functions on $\mathbb{R},$ since sums, additive inverses, and products of continuous functions are continuous.

Ideals

29.Ideals are the subsets that allow the construction of quotient rings, playing a role analogous to normal subgroups in group theory. A nonempty subset $I \subseteq R$ is called a left ideal of $R$ when it is closed under addition and under left multiplication by elements of $R,$ that is, for all $a \in I$ and $r \in R$ the element $r \cdot a$ belongs to $I.$ A right ideal is defined analogously with right multiplication. A subset that is simultaneously a left and a right ideal is called a two-sided ideal, or simply an ideal.

30.The set $n\mathbb{Z}$ of all multiples of a fixed integer $n$ is an ideal of $\mathbb{Z}$: for any $a = nk \in n\mathbb{Z}$ and any $r \in \mathbb{Z}$:

$$r \cdot a = n(rk) \in n\mathbb{Z}$$

31.

Ideals are precisely the kernels of ring homomorphisms, a fact that makes them the natural tool for constructing quotient rings and for studying the structure of rings through their homomorphic images.

Ring homomorphisms and isomorphisms

32.A ring homomorphism is a function between two rings that preserves both operations. Given two rings $(R, +, \cdot)$ and $(S, \oplus, \odot),$ a function $\varphi : R \to S$ is a ring homomorphism when for all $a, b \in R$:

$$\varphi(a + b) = \varphi(a) \oplus \varphi(b)$$

$$\varphi(a \cdot b) = \varphi(a) \odot \varphi(b)$$

33.The first condition requires that $\varphi$ be a group homomorphism between the additive groups, and the second that it preserve multiplication. As a consequence, $\varphi$ maps the additive identity of $R$ to the additive identity of $S.$ When both rings are unital, one often additionally requires that $\varphi(1_R) = 1_S.$ The kernel and image of a ring homomorphism $\varphi : R \to S$ are defined as in the case of groups:

$$\ker(\varphi) = \{\ a \in R : \varphi(a) = 0_S \ \}$$

$$\mathrm{im}(\varphi) = \{\ \varphi(a) : a \in R \ \}$$

34.The kernel is always an ideal of $R,$ and the image is always a subring of $S.$ A homomorphism is injective if and only if its kernel contains only the additive identity of $R.$

35.Polynomial evaluation at a matrix gives another example. Fix a matrix $T \in \mathrm{M}_n(\mathbb{R})$ and define the following map.

$$\mathrm{ev}_T : \mathbb{R}[x] \to \mathrm{M}_n(\mathbb{R})$$

$$a_0 + a_1x + \cdots + a_mx^m \mapsto a_0I_n + a_1T + \cdots + a_mT^m$$

36.For polynomials $p$ and $q,$ the identities $(p + q)(T) = p(T) + q(T)$ and $(pq)(T) = p(T)q(T)$ follow from the laws of matrix multiplication and the rule $T^iT^j = T^{i+j}.$ Hence $\mathrm{ev}_T$ is a ring homomorphism. When $n = 1,$ this construction reduces to ordinary evaluation at a real number.

37.A ring homomorphism that is both injective and surjective is called a ring isomorphism. Two rings are isomorphic, written $R \cong S,$ when an isomorphism between them exists. Isomorphic rings are structurally identical and share all properties that are intrinsic to their ring structure.

38.As an example, consider the map $\varphi : \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ defined by $\varphi(a) = a \bmod n.$ This map preserves both operations, since the residue class of a sum coincides with the sum of the residue classes in $\mathbb{Z}/n\mathbb{Z},$ and analogously for the product. It is therefore a ring homomorphism, and its kernel is precisely $n\mathbb{Z},$ the ideal of multiples of $n.$

39.Two rings $R$ and $S$ determine a ring structure on the Cartesian product $R \times S$ by componentwise operations:

$$(r,s) + (r',s') = (r + r',s + s')$$

$$(r,s)(r',s') = (rr',ss')$$

40.For two factors this ring is also denoted by $R \oplus S.$ If $a$ and $b$ are relatively prime positive integers, consider the following map.

$$ \Phi : \mathbb{Z}/ab\mathbb{Z} \to \mathbb{Z}/a\mathbb{Z} \oplus \mathbb{Z}/b\mathbb{Z}$$

$$[x]_{ab} \mapsto ([x]_a,[x]_b)$$

41.This map is a ring isomorphism. It is well-defined because $x \equiv y \pmod{ab}$ implies both $x \equiv y \pmod a$ and $x \equiv y \pmod b,$ and the componentwise operations show that it preserves sums and products. If $\Phi([x]_{ab}) = ([0]_a,[0]_b),$ then both $a$ and $b$ divide $x.$ Since $\gcd(a,b) = 1,$ their product divides $x,$ so $[x]_{ab} = [0]_{ab}$ and $\Phi$ is injective. The Chinese remainder theorem supplies, for every pair $([r]_a,[s]_b),$ an integer $x$ satisfying $x \equiv r \pmod a$ and $x \equiv s \pmod b.$ Thus $\Phi$ is surjective. This formulation expresses simultaneous congruences as an isomorphism between rings.

42.

Adding the requirements that $1 \neq 0$ and every nonzero element be invertible promotes a commutative ring with unity to a field, the structure in which linear algebra is developed and on which the theory of vector spaces is built. The general framework of structure-preserving maps across the standard algebraic structures is treated on the page about homomorphisms and isomorphisms.