87 articles
Prime element
A prime element is a nonzero, noninvertible element of a commutative ring such that when it divides a product ab, it divides a or b, the ring-theoretic generalization of ordinary prime numbers.
Prime ideal
A prime ideal is a proper ideal in a commutative ring such that whenever a product ab lies in it, one factor lies in it.
Principal ideal domain
A principal ideal domain (PID) is an integral domain in which every ideal is generated by a single element, examples including the integers and the Gaussian integers.
Projective module
A projective module, also called a finitely generated projective module in the finite case, is a module over a ring that behaves like a free module, characterized by the lifting property for homomorphisms.
Puiseux series
A Puiseux series is a power series in fractional powers of a variable that expresses the local behavior of algebraic functions; the method goes back to Newton.
Quotient ring
A quotient ring, also called a factor ring or residue class ring, is a ring built from a ring R and an ideal I, generalizing modular arithmetic.
Radical of an ideal
The radical of an ideal I in a commutative ring, written √I, is the set of elements some positive power of which lies in I; it is itself an ideal containing I.
Rational root theorem
The rational root theorem states that any rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient.
Reduced ring
A reduced ring is a ring containing no nonzero nilpotent elements, so x squared equaling zero forces x to be zero; examples include every integral domain and Boolean ring.
Regular local ring
A regular local ring is a commutative Noetherian local ring whose maximal ideal can be generated by as many elements as its Krull dimension, making it algebraically smooth.
Regular sequence
A regular sequence is a sequence of ring elements where each is a nonzerodivisor on the module left by the earlier ones, used to define depth.
Ring (mathematics)
In mathematics, a ring is a set with addition and multiplication forming an Abelian group under addition and linked by distributive laws; the integers are a familiar example.
Ring homomorphism
A ring homomorphism, also called a ring morphism, is a function between rings that preserves addition and multiplication, with kernel and image governed by the isomorphism theorems.
Ring of integers
The ring of integers of a number field K is the subring of algebraic integers in K, generalizing the ordinary integers and central to algebraic number theory.
Ring theory
Ring theory is the branch of algebra that studies rings, sets with addition and multiplication obeying basic axioms, with commutative and noncommutative branches and examples like the integers.
Semigroup algebra
A semigroup algebra, also called a semigroup ring, is a ring of finite formal sums of semigroup elements multiplied by the semigroup operation, generalizing the polynomial ring.
Semiring
A semiring is an algebraic structure with addition and multiplication in which multiplication distributes over addition, but subtraction is not required; the natural numbers are the basic example.
Serre's multiplicity conjectures
Serre's multiplicity conjectures are statements in commutative algebra about Serre's Tor-formula intersection multiplicity, defined in the late 1950s; nonnegativity and vanishing are proved, positivity remains open.
Skew polynomial ring
A skew polynomial ring, also called an Ore extension, is a noncommutative polynomial ring in which the variable does not commute with coefficients, introduced by Øystein Ore in 1933.
Stark–Heegner theorem
The Stark–Heegner theorem, also called the Baker–Heegner–Stark theorem, states that exactly nine imaginary quadratic fields have class number one; Heegner proved it in 1952, and Baker and Stark independently in 1966 and 1967.
Structure theorem for finitely generated modules over a principal ideal domain
The structure theorem for finitely generated modules over a principal ideal domain classifies every such module as a product of cyclic modules, generalizing the classification of finitely generated abelian groups.
Unique factorization domain
A unique factorization domain, also called a factorial ring or factorial domain, is an integral domain in which every nonzero nonunit factors uniquely into irreducible elements.
Unit (ring theory)
In ring theory, a unit is an element of a ring that has a multiplicative inverse, and the units together form a group under multiplication.
Valuation ring
A valuation ring is an integral domain in which, for every pair of elements, one divides the other; it carries a valuation and is a local ring.
Weyl algebra
The Weyl algebra, denoted A_n(K), is the non-commutative algebra of differential operators with polynomial coefficients, named for Hermann Weyl, central to D-module theory and quantum mechanics.
Witt vectors
Witt vectors are a construction turning a commutative ring R into a ring W(R) of sequences, built from universal integer polynomials; for finite fields they yield p-adic integers.
Zero divisor
A zero divisor is a nonzero element of a ring that gives zero when multiplied by another nonzero element, as 2 · 3 = 0 in the integers modulo 6.