one of the basic concepts of modern algebra. The systems (sets) of numbers listed below, together with the operations of addition and multiplication, are the simplest examples of rings: (1) the set of all positive and negative integers and zero; (2) the set of all even integers, or more generally, the set of all multiples of a given integer n; (3) the set of all rational numbers. A common feature of these three examples is that addition and multiplication (as well as subtraction) of the numbers within each system does not lead to numbers outside the system. In different branches of mathematics it is often necessary to deal with various sets (consisting, perhaps, of polynomials or matrices; see examples 7 and 9 below), on whose elements two operations are carried out that are very similar in their properties to the addition and multiplication of ordinary numbers. The study of the properties of a broad class of such sets in the subject of ring theory.
A ring is a nonempty set R on whose elements two operations are defined—addition and multiplication—which associate to any two elements a and b in R, taken in a definite order, a single element a +b in R—their sum—and a single element ab in R—their product—such that the following conditions (ring axioms) hold:
I. Commutativity of addition.
a + b = b + a
II. Associativity of addition:
a + (b + c) = (a + b) + c
III. Invertibility of addition (possibility of subtraction): the equation a + x = b admits the solution x = b — a
a(b + c) = ab + ac (b + c)a = ba + ca
The properties enumerated above show that the elements of a ring form a commutative group with respect to addition. The following sets are further examples of rings: (4) all real numbers; (5) all complex numbers; (6) complex numbers of the form a + bi with integral a and b; (7) polynomials in one variable x with rational, real, or complex coefficients; (8) all functions that are continuous on a given interval of the real number line; (9) all square matrices of order n with real (or complex) elements; (10) all quaternions; (11) all Cayley-Dickson numbers, that is, expressions of the form α + βe, where α and β are quaternions and e is a letter and addition and multiplication and defined by the equalities (α + βe) + (α1 + β1e) = (α + α1) + (β + β1)e and (α + βe) (α1 +β1e) = αα + ββ1) + (αα + βᾱ1)e, where ᾱ is the quaternion that is conjugate to α; (12) all symmetric matrices of order n with real elements with respect to matrix addition and “Jordan multiplication” a.b = ½(ab + ba); (13) vectors in three-dimensional space under ordinary addition and vector multiplication.
In many cases, additonal restrictions are placed upon the multiplication operation in rings. Thus, if a(bc) = (ab)c, then the ring is called associative (examples 1–10); if the equalities (aa)b = a(ab) and (ab)b = a(bb) are satisfied in a ring, then it is called an alternative ring (example 11); if the equalities ab = ba and (ab)(aa) = ((aa)b)a are satisfied in a ring, then it is called a Jordan ring (example 12); if the equalities a(bc) + b(ca) + c(ab) = 0 and a2 = 0 are satisfied in a ring, then it is called a Lie ring (example 13); and if ab = ba, then the ring is called commutative (examples 1–8, 12).
The operations of addition and multiplication in rings are similar in their properties to the corresponding operations on numbers. Thus, the elements of a ring may be subtracted as well as added; there exists an element 0 (zero) with the usual properties; for any element a there exists an additive inverse, that is, an element -a such that a +(-a) = 0; the product of any element with the element 0 is always equal to zero. However, from examples 8, 9, 12, and 13 it can be seen a ring may contain nonzero elements a and b whose product is equal to zero; ab = 0. Such elements are called zero divisors. An associative commutative ring with no zero divisors is called an integral domain (examples 1–7). Just as in the domain of integers, it is not always possible in every ring to divide one element by another. However, if this is possible, that is, if the equations ax = b and ya = b are always solvable for a ≠ 0, then the ring is called a division ring (examples 3–5, 10, 11). An associative commutative division ring is called a field (examples 3–5). Polynomial rings in one or more variables over an arbitrary field and rings of matrices over associative division rings, defined analogously to the ring of examples 7 and 9, are very important in many branches of algebra.
Many classes of rings are increasingly finding applications outside algebra. The most important are rings of functions and rings of operators, which have played a large role in the development of functional analysis; alternative division rings, which are applicable in projective geometry; and differential rings and fields, which reflect an interesting attempt to apply ring theory to differential equations.
One of the most fruitful techniques in the study of rings is the use of homomorphic mappings, or homomorphisms. A homo-morphism is a single-valued mapping R → R’ of the ring R into the ring R’ such that a → a’ and b → b’ imply a + b → a’ + b’ and ab → a’b’. If this mapping is also one-to-one, then it is called an isomorphism and the rings R and R’ are said to be isomorphic. Isomorphic rings possess identical algebraic properties.
A set M of the elements of a ring R is called a subring if M is itself a ring with respect to the operations defined in R. A subring M is called a left (right, two-sided) ideal of the ring R if for any elements m in M and r in R the product rm (mr rm and mr) belongs to M. Elements a and b of the ring R are called congruent with respect to the ideal M if a — b belongs to M. The entire ring is partitioned into classes of congruent elements— residue classes with respect to the ideal M. If the addition and multiplication of the residue classes with respect to a two-sided ideal M is defined by means of the addition and multiplication of the elements of these classes, then these residue classes themselves form a ring—the factor (or difference) ring R/M of the ring R by the ideal M. An important theorem is the theorem on the homomorphism of rings, which states that if we associate to every element of a ring its residue class, then we obtain a homomorphic mapping of the ring R into the factor ring R/M; conversely, if R is mapped homomorphically into R’, then the set M of the elements of R that are mapped into the zero of the ring R’ is a two-sided ideal in R, and R’ is isomorphic to R/M.
Of the various types of rings, the so-called algebras are easiest to study and to apply. A ring R is called an algebra over a field P if for any a in P and r in R the product αr is also in R, and moreover (α +β)r = αr +βr, α(r + s) = αr +αs, (αβ)r = α(βr), α(rs) = (αr)s = r(αs),єr = r for all α,β in P and r, s in R, where є is the multiplicative identity of the field P. If all elements of an algebra can be expressed as linear combinations of n linearly independent elements, then R is called an algebra of rank n, or a hypercomplex system. Examples of algebras are the complex numbers (an algebra of rank 2 over the real number field), the complete ring of matrices with elements in the field P (an algebra of rank n2 over P), the ring of example 10 (an algebra of rank 4 over the real number field), and the ring of example 8.
For the integers and for polynomial rings, we have a theorem that asserts the possibility of unique decomposition of an element into a product of primes, that is, nondecomposable elements. This theorem holds for any principal ideal domain, that is, an integral domain in which any ideal consists of multiples of a single element. A frequent case of such a domain is a Euclidean ring, that is, a ring in which to each element a ≠ 0 there corresponds a nonnegative integer n(a) such that n(ab) ≥ n(a), and for any a and b ≠ 0 there exist q and r such that a = bq + r and either n(r) < n(b) or r = 0. Examples of Euclidean rings are polynomial rings and the rings of examples 1 and 6. For a broad class of rings, a theorem concerning the unique decomposition of an ideal into a product of prime ideals is valid, although the corresponding theorem does not hold for the ring elements themselves. The foundations of the theory of decomposition of ideals in abstract rings were laid by E. Noether in the 1920’s.
One of the first to study ring theory in Russia was E. I. Zolotarev (1870’s); his research pertained to number rings (more specifically, to the decomposition of ideals in such rings). In the Soviet Union, ring theory is primarily being worked on at three research centers: Moscow, Novosibirsk, and Kishinev.
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Entsiklopediia elementarnoi matematiki, book 1. Moscow-Leningrad, 1951.
Van der Waerden, B. L. Sovremennaia algebra, parts 1–2. Moscow-Leningrad, 1947. (Translated from German.)
Jacobson, N. Stroenie kolets. Moscow, 1961. (Translated from English.)
Lang, S. Algebra. Moscow, 1968. (Translated from English.)