Ring (mathematics)

A ring is a set equipped with two operations, addition and multiplication, whose laws generalize the arithmetic of integers. Addition forms an abelian group, multiplication is associative, and multiplication distributes over addition. Multiplication need not be commutative, and nonzero elements need not have multiplicative inverses. Rings are fundamental objects of abstract algebra, encompassing number systems, polynomial expressions, and matrices within one framework. Conventions differ over whether a multiplicative identity is required; this entry assumes one unless stated otherwise. (math.ucla.edu)

Definition and conventions

A ring $R$ has two binary operations, usually written $a+b$ and $ab$, defined for every pair of elements. Its axioms require:

• Abelian addition: addition is associative and commutative; there is an additive identity $0$; and every $a$ has an additive inverse $-a$.
• Associative multiplication: $(ab)c=a(bc)$.
• Distributivity: $a(b+c)=ab+ac$ and $(a+b)c=ac+bc$.
• Multiplicative identity: an element $1$ satisfies $1a=a1=a$.

The additive structure is therefore an abelian group, a central concept in group theory. These axioms imply $0a=a0=0$ and $(-a)b=-(ab)$. A ring is commutative when $ab=ba$ for all elements. (math.ucla.edu)

Some authors omit the identity axiom and distinguish rings “with identity” or “with unity.” Under that convention, the even integers $2\mathbb Z$, with ordinary operations, form a ring without a multiplicative identity. The zero ring, consisting of one element, satisfies the unital axioms with $0=1$; definitions of fields and integral domains exclude it. A subring may likewise be required to share the identity of its parent ring, depending on convention. (faculty.niu.edu)

Examples

The integers $\mathbb Z$ form a commutative ring, but division by a nonzero integer does not generally remain within $\mathbb Z$. The rational numbers, real numbers, and complex numbers are rings with the stronger structure of a field: every nonzero element has a multiplicative inverse. (joemileti.site)

Several constructions supply other important examples:

• Residue rings: $\mathbb Z/n\mathbb Z$ consists of integer congruence classes, with operations performed using modular arithmetic.
• Polynomial rings: $R[x]$ consists of polynomials in an indeterminate $x$, with coefficients in $R$. If $R$ is commutative, so is $R[x]$.
• Matrix rings: $M_n(R)$ consists of square matrices, using matrix addition and multiplication. Over a nonzero unital ring, multiplication is generally noncommutative when $n\geq2$.
• Function rings: real-valued functions on a fixed set form a commutative ring under pointwise addition and multiplication.

These examples show that ring elements need not be numbers and that multiplication need not resemble ordinary numerical multiplication. (math.ucla.edu)

Units, zero divisors, and domains

A unit is an element $u$ possessing a two-sided multiplicative inverse: $uv=vu=1$ for some $v$. Units form a group under multiplication. In $\mathbb Z$, the only units are $1$ and $-1$. (joemileti.site)

In a commutative ring, a zero divisor is a nonzero element $a$ for which $ab=0$ with some nonzero $b$. For example, in $\mathbb Z/6\mathbb Z$, the nonzero classes of $2$ and $3$ multiply to zero. Thus, a product can vanish without either factor vanishing. (joemileti.site)

An integral domain is a nonzero commutative unital ring without zero divisors. Multiplication by a nonzero element then satisfies cancellation. Every field is an integral domain, but $\mathbb Z$ shows that the converse fails. Every finite integral domain is a field; consequently, $\mathbb Z/n\mathbb Z$, for $n\geq2$, is a field exactly when $n$ is a prime number. Every integral domain embeds in a field of fractions, generalizing the construction of rational numbers from integers. (faculty.niu.edu)

Ideals, quotients, and homomorphisms

An ideal is an additive subgroup that absorbs multiplication by ring elements. In a noncommutative ring, left, right, and two-sided ideals must be distinguished. A two-sided ideal $I$ permits the construction of the quotient ring $R/I$, whose elements are additive cosets. Its operations are

The absorption property makes these operations independent of the chosen representatives. (crypto.stanford.edu)

A ring homomorphism preserves addition and multiplication and, under the unital convention, preserves $1$. Its kernel is a two-sided ideal. The first isomorphism theorem gives an isomorphism

Thus quotients describe the identifications made by structure-preserving maps. (crypto.stanford.edu)

For commutative unital rings, a proper ideal $I$ is a prime ideal precisely when $R/I$ is an integral domain, and a maximal ideal precisely when $R/I$ is a field. (faculty.niu.edu)

Structural theory and geometry

Ring theory extends linear algebra through modules, which resemble vector spaces but use scalars from a ring rather than a field. A commutative ring is Noetherian when every ideal is finitely generated, equivalently when every ascending chain of ideals stabilizes. Hilbert’s basis theorem states that $R[x]$ is Noetherian whenever $R$ is, providing finiteness results for polynomial equations. (jmilne.org)

In algebraic geometry, a quotient $k[x_1,\ldots,x_n]/I$ records polynomial expressions subject to the relations in $I$. Prime ideals also form the spectrum of a commutative ring, a space equipped with the Zariski topology. This construction underlies schemes and connects algebraic properties of rings with geometric properties of spaces. (jmilne.org)