GLOSSARY OF RING THEORY

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

Contents

Definition of a ring

Types of elements

Homomorphisms and ideals

Types of rings

Miscellaneous

Definition of a ring

;'Ring' : A ''ring'' is a set ''R'' with two binary operations, usually called addition (+) and multiplication (
★ ), such that ''R'' is an abelian group under addition, a monoid under multiplication, and such that multiplication is both left and right distributive over addition. Note that rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1.
; 'Subring' : A subset ''S'' of the ring (''R'',+,
★ ) which remains a ring when + and
★ are restricted to ''S'' and contains the multiplicative identity 1 of ''R'' is called a ''subring'' of ''R''.

Types of elements

; 'Central' : An element ''r'' of a ring ''R'' is ''central'' if ''xr'' = ''rx'' for all ''x'' in ''R''. The set of all central elements forms a subring of ''R'', known as the ''center'' of ''R''.
; 'Divisor' : In an integral domain ''R'', an element ''a'' is called a ''divisor'' of the element ''b'' (and we say ''a'' ''divides'' ''b'') if there exists an element ''x'' in ''R'' with ''ax'' = ''b''.
; 'Idempotent' : An element ''e'' of a ring is ''idempotent'' if ''e''² = ''e''.
; 'Unit' or 'invertible element' : An element ''r'' of the ring ''R'' is a ''unit'' if there exists an element ''r''^-1 such that ''rr''^-1=''r''^-1''r''=1. This element ''r''^-1 is uniquely determined by ''r'' and is called the ''multiplicative inverse'' of ''r''. The set of units forms a group under multiplication.
; 'Irreducible' : An element ''x'' of an integral domain is ''irreducible'' if it is not a unit and for any elements ''a'' and ''b'' such that ''x''=''ab'', either ''a'' or ''b'' is a unit. Note that every prime element is irreducible, but not necessarily vice versa.
; 'Prime element' : An element ''x'' of an integral domain is a ''prime element'' if it is not zero and not a unit and whenever ''x'' divides a product ''ab'', ''x'' divides ''a'' or ''x'' divides ''b''.
; 'Nilpotent' : An element ''r'' of ''R'' is ''nilpotent'' if there exists a positive integer ''n'' such that ''r''^''n'' = 0.
; 'Zero divisor' : A nonzero element ''r'' of ''R'' is said to be a ''zero divisor'' if there exists a nonzero element ''s'' in R such that ''sr''=0 or ''rs''=0. If a ring has a zero divisor which is also a unit, then the ring has no other elements and is the trivial ring {0}.

Homomorphisms and ideals

; 'Factor ring' : Given a ring ''R'' and an ideal ''I'' of ''R'', the ''factor ring'' is the set ''R''/''I'' of cosets {''a+I'' : ''a''∈''R''} together with the operations (''a+I'')+(''b+I'')=(''a''+''b'')+''I'' and (''a+I'')
★ (''b+I'')=''ab+I'', which turn ''R''/''I'' into a ring. The relationship between ideals, homomorphisms, and factor rings is summed up in the fundamental theorem on homomorphisms.
; 'Finitely generated ideal' : A left ideal ''I'' is ''finitely generated'' if there exist finitely many elements ''a''₁,...,''a''_''n'' such that ''I'' = ''Ra''₁ + ... + ''Ra''_''n''. A right ideal ''I'' is ''finitely generated'' if there exist finitely many elements ''a''₁,...,''a''_''n'' such that ''I'' = ''a''₁''R'' + ... + ''a''_''n''''R''. A two-sided ideal ''I'' is ''finitely generated'' if there exist finitely many elements ''a''₁,...,''a''_''n'' such that ''I'' = ''Ra''₁''R'' + ... + ''Ra''_''n''''R''.
; 'Ideal' : A ''left ideal'' ''I'' of ''R'' is a subgroup of (''R'',+) such that ''aI'' ⊆ ''I'' for all ''a''∈''R''. A ''right ideal'' is a subgroup of (''R'',+) such that ''Ia''⊆''I'' for all ''a''∈''R''. An ''ideal'' (sometimes for emphasis: a ''two-sided ideal'') is a subgroup which is both a left ideal and a right ideal.
; 'Jacobson radical' : The intersection of all maximal left ideals in a ring forms a two-sided ideal, the ''Jacobson radical'' of the ring.
; 'Kernel of a ring homomorphism' : The ''kernel'' of a ring homomorphism ''f'' : ''R'' → ''S'' is the set of all elements ''x'' of ''R'' such that ''f''(''x'') = 0. Every ideal is the kernel of a ring homomorphism and vice versa.
; 'Maximal ideal' : A left ideal ''M'' of the ring ''R'' is a ''maximal left ideal'' if ''M'' ≠ ''R'' and the only left ideals containing ''M'' are ''R'' and ''M'' itself. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of ''maximal ideals''.
; 'Nil ideal' : An ideal is ''nil'' if it consists only of nilpotent elements.
; 'Nilpotent ideal' : An ideal ''I'' is ''nilpotent'' if its powers ''I''^''k'' are {0} for ''k'' large enough. Every nilpotent ideal is nil, but the converse is not true in general.
; 'Nilradical' : The set of all nilpotent elements in a commutative ring forms an ideal, the ''nilradical'' of the ring. The nilradical is equal to the intersection of all the ring's prime ideals. It is contained in, but in general not equal to, the ring's Jacobson Radical.
; 'Prime ideal' : An ideal ''P'' in a commutative ring ''R'' is ''prime'' if ''P'' ≠ ''R'' and if for all ''a'' and ''b'' in ''R'' with ''ab'' in ''P'', we have ''a'' in ''P'' or ''b'' in ''P''. Every maximal ideal in a commutative ring is prime. There is also a definition of prime ideal for noncommutative rings.
; 'Principal ideal' : a ''principal left ideal'' in the ring ''R'' is a left ideal of the form ''Ra'' for some element ''a'' of ''R''; a ''principal right ideal'' is a right ideal of the form ''aR'' for some element ''a'' of ''R''; a ''principal ideal'' is a two-sided ideal of the form ''RaR'' for some element ''a'' of ''R''.
; 'Radical of an ideal' : The radical of an ideal ''I'' in a commutative ring consists of all those ring elements a power of which lies in ''I''. It is equal to the intersection of all prime ideals containing ''I''.
; 'Ring homomorphism' : A function ''f'' : ''R'' → ''S'' between rings (''R'',+,
★ ) and (''S'',⊕,×) is a ''ring homomorphism'' if it satisfies
:: ''f''(''a'' + ''b'') = ''f''(''a'') ⊕ ''f''(''b'')
:: ''f''(''a''
★ ''b'') = ''f''(''a'') × ''f''(''b'')
:: ''f''(1) = 1
:for all elements ''a'' and ''b'' of ''R''.
; 'Ring monomorphism' : A ring homomorphism that is injective is a ''ring monomorphism''.
; 'Ring isomorphism' : A ring homomorphism that is bijective is a ''ring isomorphism''. The inverse of an isomorphism, it turns out, is also a ring isomorphism. Two rings are ''isomorphic'' if there exists a ring isomorphism between them. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.

Types of rings

; 'Artinian ring' : A ring satisfying the descending chain condition for left ideals is ''left artinian''; if it satisfies the descending chain condition for right ideals, it is ''right artinian''; if it is both left and right artinian, it is called ''artinian''. Artinian rings are noetherian.
; 'Boolean ring' : A ring in which every element is idempotent is a ''boolean ring''.
; 'Commutative ring' : A ring ''R'' is ''commutative'' if the multiplication is commutative, i.e. ''rs''=''sr'' for all ''r'',''s''∈''R''.
; 'Dedekind domain' : A ''Dedekind domain'' is an integral domain in which every ideal has a unique factorization into prime ideals.
; 'Division ring' or 'skew field' : A ring in which every nonzero element is a unit and 1≠0 is a ''division ring''.
; 'Domain (ring theory)' : A ''domain'' is a ring without zero divisors and in which 1≠0. This is the noncommutative generalization of integral domain.
; 'Euclidean domain' : An integral domain in which a degree function is defined so that "division with remainder" can be carried out is called a ''Euclidean domain'' (because the Euclidean algorithm works in these rings). All Euclidean domains are principal ideal domains.
; 'Field' : A commutative division ring is a ''field''. Every finite division ring is a field, as is every finite integral domain. Field theory is in fact an older branch of mathematics than ring theory.
; 'Integral domain' or 'entire ring' : A commutative ring without zero divisors and in which 1≠0 is an ''integral domain''.
;'Invariant basis number': A ring ''R'' has ''invariant basis number'' if ''R''^''m'' isomorphic to ''R''^''n'' as ''R''-modules implies ''m''=''n''.
; 'Local ring' : A ring with a unique maximal left ideal is a ''local ring''. These rings also have a unique maximal right ideal, and the left and the right unique maximal ideals coincide. Any ring can be made local via localization.
; 'Noetherian ring' : A ring satisfying the ascending chain condition for left ideals is ''left noetherian''; a ring satisfying the ascending chain condition for right ideals is ''right noetherian''; a ring that is both left and right noetherian is ''noetherian''. A ring is left noetherian if and only if all its left ideals are finitely generated; analogously for right noetherian rings.
; 'Prime ring' : A non-trivial ring ''R'' is called a ''prime ring'' if for any two elements ''a'' and ''b'' of ''R'' with ''aRb'' = 0, we have either ''a = 0'' or ''b = 0''. This is equivalent to saying that the zero ideal is a prime ideal. Every simple ring and every domain is a prime ring.
; 'Primitive ring' : A ''left primitive ring'' is a ring that has a faithful simple left ''R''-module. Every simple ring is primitive. Primitive rings are prime.
; 'Semisimple ring' : A ''semisimple ring'' is a ring ''R'' that has a "nice" decomposition, in the sense that ''R'' is a semisimple left ''R''-module. Every semisimple ring is also Noetherian, and has no nilpotent ideals. Any ring can be made semi-simple if it is divided by its Jacobson radical.
; 'Simple ring' : A non-zero ring with no non-zero two-sided ideals is a ''simple ring''.
; 'Trivial ring or zero ring' : The ring consisting only of a single element 0=1.
; 'Unique factorization domain' or 'factorial ring': An integral domain ''R'' in which every non-zero non-unit element can be written as a product of prime elements of ''R''. This essentially means that every non-zero non-unit can be written uniquely as a product of irreducible elements.
; 'Principal ideal domain' : An integral domain in which every ideal is principal is a ''principal ideal domain''. All principal ideal domains are unique factorization domains.

Miscellaneous

; 'Characteristic' : The ''characteristic'' of a ring is the smallest positive integer ''n'' satisfying ''nx''=0 for all elements ''x'' of the ring.
; 'Direct product' of a family of rings : This is a way to construct a new ring from given rings by taking the cartesian product of the given rings and defining the algebraic operation component-wise.
; 'Krull dimension of a commutative ring' : The maximal length of a strictly increasing chain of prime ideals in the ring.
; 'Localization of a ring' : A technique to turn a given set of elements of a ring into units. It is named ''Localization'' because it can be used to make any given ring into a ''local'' ring. To localize a ring ''R'', take a multiplicatively closed subset ''S'' containing no zero divisors, and formally define their multiplicative inverses, which shall be added into ''R''.
; 'Rng' : A rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" is meant to suggest that it is a "r'i'ng" without an "'i'dentity".
; 'Semiring' : A ''semiring'' is an algebraic structure satisfying the same properties as a ring, except that addition need only be an abelian monoid operation, rather than an abelian group. That is, elements in a semiring need not have additive inverses.