Rings and algebras

Sets with two binary operations, usually called addition and multiplication. Such a set with an addition and a multiplication is called a ring if: 1) it is an Abelian group with respect to addition (in particular, the ring has a zero element, denoted by 0, and a negative element $ - x $ for each element $ x $); and 2) the multiplication operation is right and left distributive with respect to addition, that is, $ x ( y + z ) = x y + x z $ and $ ( y + z ) x = yx + z x $, for all $ x , y , z $ in the ring.

If the ring $ K $ has no divisors of zero, that is, if $ x y \neq 0 $ for any non-zero $ x , y \in K $, then the set of all non-zero elements of the ring is a groupoid with respect to multiplication. The ring is a skew-field if the set of all non-zero elements forms a group with respect to multiplication. A ring $ K $ is called associative if the multiplication in it satisfies the associative law, that is, $ ( x y ) z = x ( y z ) $ for all $ x , y , z $ in $ K $. If the multiplication in the ring is commutative, that is, $ xy = yx $ for all $ x, y $ in $ K $, then the ring is called commutative. By an identity is meant an element 1 of the ring such that

$$ x \cdot 1 = 1 \cdot x = x $$

for all $ x \in K $. In general, a ring need not have an identity. Every skew-field is an associative ring with an identity and without divisors of zero. A commutative associative ring without divisors of zero and with an identity is called an integral domain.

Let $ \Phi $ be an associative ring with identity 1. Then a (not necessarily associative) ring $ A $ is called an algebra over $ \Phi $, or a ring with operators $ \Phi $, if for any two elements $ \alpha \in \Phi $, $ a \in A $ there is a unique product $ \alpha a \in A $ such that the following relations hold for all $ \alpha , \beta \in \Phi $, $ a , b \in A $:

$$ \tag{1 } \left . \begin{array}{cc} ( \alpha + \beta ) a = \alpha a + \beta \alpha , &\alpha ( a + b ) = \alpha a + \alpha b , \\ \alpha ( \beta a ) = ( \alpha \beta ) a , &1 a = a ,\ \alpha ( a b ) = ( \alpha a ) b . \\ \end{array} \right \} $$

If $ \Phi $ is commutative, then it is customary to require that the last of conditions (1) be strengthened:

$$ \tag{2 } \alpha ( a b ) = ( \alpha a ) b = a ( \alpha b ) . $$

Any ring can be regarded as an algebra over the ring of the integers by taking the product $ n a $ (where $ n $ is an integer) to be the usual one, that is, $ a + \dots + a $ ($ n $ times). Therefore a ring can be regarded as a special case of an algebra.

If $ A $ is an algebra over a field $ \Phi $, then, by definition, $ A $ is a vector space over $ \Phi $ and therefore has a basis. This makes it possible to construct an algebra over a field in terms of its basis, for which it suffices to define the multiplication table of the basis elements. An algebra over a field is called finite dimensional if it has a finite basis, that is, if it is finite dimensional as a vector space over the field.

The best-known examples of algebras are algebras of square matrices, algebras of polynomials and algebras of formal power series over fields.

An important role is played in the theory of rings and algebras, as in any other algebraic theory, by the notions of homomorphism and isomorphism. Many arguments and descriptions are carried out "up to isomorphism" , that is, isomorphic rings and algebras are not distinguished. The notion of a homomorphism is closely related to those of an ideal and a subalgebra (subring).

Let $ A $ and $ B $ be two algebras (over some fixed ring $ \Phi $ with an identity). A mapping $ \phi : A \rightarrow B $ of the set $ A $ into the set $ B $ is called a homomorphism of the algebra $ A $ into the algebra $ B $ if it "preserves the operations of the algebra" , that is, if

$$ \phi ( x + y ) = \phi ( x) + \phi ( y) , $$

$$ \phi ( x y ) = \phi ( x) \phi ( y) , $$

$$ \phi ( \alpha x ) = \alpha \phi ( x) $$

for any $ x , y \in A $, $ \alpha \in \Phi $. A homomorphism $ \phi $ is called an isomorphism if $ \phi $ is a one-to-one mapping from $ A $ onto $ B $. The latter is equivalent to saying that the image of the homomorphism $ \phi $,

$$ \mathop{\rm Im} \phi = \phi ( A) = \{ {\phi ( a) } : {a \in A } \} , $$

which is in general a subalgebra of $ B $, coincides with the whole of $ B $, while the kernel of the homomorphism $ \phi $,

$$ \mathop{\rm Ker} \phi = \{ {a \in A } : {\phi ( a) = 0 } \} , $$

which is in general a two-sided ideal of $ A $, is in this case the zero ideal. Only the two-sided ideals of an algebra $ A $ are kernels of homomorphisms from this algebra, while the homomorphic images of $ A $, up to isomorphism, are accounted for by all quotient algebras of $ A $ by all possible two-sided ideals of it.

Passing from an algebra to its subalgebras and its homomorphic images is one of the methods for obtaining new algebras. For example, one can obtain any commutative associative algebra over a field $ \Phi $ as a homomorphic image of an algebra of polynomials (in a sufficient number of variables) over $ \Phi $. Other frequently-applied constructions that should be mentioned are direct sums and direct and subdirect products of rings and algebras.

Historical information.[edit]

Until about the middle of the 19th century only individual examples of rings were known: number rings, that is, subrings of the field of complex numbers, emerging in connection with the requirements of the theory of algebraic equations, and residue-class rings of integers in number theory. The general concept of a ring did not exist.

The first examples of non-commutative rings and algebras are encountered (1843–1844) in the work of W.R. Hamilton and H. Grassmann. These are the skew-field of quaternions (cf. Quaternion), the algebra of biquaternions and the exterior algebra. The concept of a hypercomplex system began to be formulated, that is, in modern terminology, a finite associative algebra over the field $ \mathbf R $ of real numbers or the field $ \mathbf C $ of complex numbers. In 1870 there appeared in papers by B. Peirce the notions of an idempotent element (cf. Idempotent) and a nilpotent element; and it was proved that if not all elements of a hypercomplex system are nilpotent, then it has at least one non-zero idempotent. This result permitted the development of "the technique of idempotentstechnique of idempotents" and of "Peirce decompositionPeirce decompositions" , which are widely applied in the study of finite-dimensional algebras. After 1870 there began a more general study of hypercomplex systems. Encountered in the works of R. Dedekind is the general notion of an (associative) ring, a skew-field and an algebra over a field (a hypercomplex system), although he called a ring an order. The term "ring" was subsequently introduced by D. Hilbert. K. Weierstrass and Dedekind proved that any finite-dimensional commutative associative algebra without nilpotent elements over the field of real numbers is a direct sum of fields that are isomorphic either to $ \mathbf R $ or $ \mathbf C $. In 1878 G. Frobenius proved that the only non-commutative skew-field of finite dimension over the field of real numbers is the skew-field of quaternions.

At the beginning of the 20th century significant results were obtained in papers by S.E. Molin and E. Cartan in the theory of hypercomplex systems. At this time there was already a fairly well-developed theory of homomorphisms, the relation between them and ideals was clarified, and the notion of a direct sum of algebras had emerged. By considering finite-dimensional associative algebras over $ \mathbf C $, Molin introduced the notion of a simple algebra, and proved that the simple algebras are precisely the general matrix algebras over $ \mathbf C $. He also introduced the notion of a radical (now called a classical radical) and proved, in essence, that if the radical of an algebra is zero, then the algebra is a direct sum of simple algebras (cf. Radical of rings and algebras). These results were rediscovered by Cartan, who extended them to algebras over $ \mathbf R $.

At the beginning of the 20th century (associative and finite-dimensional) algebras over an arbitrary field began to be studied, rather than merely over the fields of real or complex numbers. J.M. Wedderburn, by perfecting Peirce's technique of idempotents, carried over the results of Molin and Cartan to the case of an arbitrary field. He also proved that any finite skew-field is commutative.

Finally, in the 1920's and 1930's the study of arbitrary associative rings and algebras began and left and right ideals of rings began to play a large part. In 1925–1926 W. Krull and E. Noether introduced and made systematic use of the maximum and minimum conditions for left ideals. In 1927 E. Artin carried over the results of Wedderburn on the decomposition of semi-simple algebras to all associative rings and algebras whose left ideals simultaneously satisfy the maximum and the minimum condition. In 1929 Noether showed that in this connection it suffices merely to require the minimum condition. In 1939 it was proved that under the minimum condition (as under the maximum condition) the radical of a ring is its largest nilpotent left ideal (see Artinian ring; Noetherian ring). Thus, by 1940 the Molin–Cartan–Wedderburn theory had been carried over to the case of associative rings and algebras with the minimum condition for their left (or right) ideals.

Basic trends in the theory of rings and algebras.[edit]

Structure theory gives a description of algebras (as a rule, satisfying certain finiteness conditions), presenting them in the form of a direct sum or a subdirect product of algebras of a simpler structure. Nowadays (that is, since about 1970) the classical Molin–Cartan–Wedderburn–Artin theory for associative rings and algebras has been carried over to the case of rings and algebras with the minimum condition for their principal left ideals. Under this condition it has in fact been proved that if the algebra does not have nilpotent ideals, then it decomposes into a direct (not necessarily finite) sum of simple algebras, while if it does not even have nilpotent elements, then into a direct sum of skew-fields. In the case when the algebra has nilpotent ideals, its structure is considerably more complicated. The best-known theorem concerning such algebras is the Wedderburn–Mal'tsev theorem "on the splitting of a radical" . It concerns the decomposition of a finite-dimensional associative algebra into the semi-direct sum of a radical and a semi-simple subalgebra. A meaningful structure theory has been created for alternative algebras (see Alternative rings and algebras), to which in fact the complete Molin–Cartan–Wedderburn–Artin theory has been carried over, as well as for Jordan algebras (cf. Jordan algebra).

A number of structure theorems have also been obtained without finiteness conditions. Already Krull proved that any commutative associative ring without nilpotent elements decomposes into a subdirect product of rings without divisors of zero. It was subsequently proved that in Krull's theorem the commutativity requirement can be dropped; thereupon a number of criteria were found for the decomposability of an arbitrary non-associative algebra into a subdirect product of algebras without divisors of zero and algebras with unique division.

The theory of simple algebras and skew-fields is closely related to structure theory, since many structure theorems reduce the study of the rings and algebras considered above to that of simple algebras and skew-fields. A description has been obtained of the associative simple algebras with an identity having minimal left ideals, and also of the finite-dimensional alternative and Jordan simple algebras. Automorphisms and derivations of simple associative algebras and skew-fields have been considered (see Algebraic system, automorphism of an; Differential algebra).

The theory of radicals is also closely related to structure theory; structure theorems are as a rule theorems concerning rings and algebras that are semi-simple in the sense of some radical. To obtain new structure theorems various different radicals have been introduced: the Baer lower nil-radical, the Levitski locally nilpotent radical, the quasi-regular Jacobson radical, the Brown–McCoy radical, etc. At the beginning of the 1950's a general theory of radicals was created that is closely related to the theory of modules and representations (see Radical of rings and algebras).

Since then, algebras with identity relations have begun to attract the attention of algebraists, since it was realized that the presence of a (non-trivial) identity strongly influences the structure of rings and algebras. In this connection there is the Kaplansky index theorem on associative algebras: If $ A $ is a primitive algebra with a polynomial identity of degree $ d $, then $ A $ is a finite-dimensional simple algebra over its centre and its dimension does not exceed $ [ d / 2 ] ^ {2} $ (see [6]). There are also several results on non-associative algebras with identity relations (see Variety of rings).

Free algebras and free products (cf. Free algebra; Free product) of algebras are important constructs in the theory of rings and algebras, since any algebra (of some variety) is a homomorphic image of the free algebra of this variety. It has been proved that any subalgebra of a free non-associative algebra is itself free, and that all subalgebras of free commutative algebras, free anti-commutative algebras and free Lie algebras are free. Investigations in this area are closely related to those of algebras with identity relations and varieties of algebras, since the identities of a given variety are defining relations in the free algebra of the given variety.

The theory of imbeddings largely studies questions of imbedding associative rings and algebras into skew-fields or simple algebras in which some or other equations are solvable (see Imbedding of rings). The example of an associative algebra without divisors of zero that is not imbeddable in a skew-field acted as a stimulus to the development of this theory. Thereupon a criterion was discovered for the existence of a (classical) quotient skew-field for associative rings and algebras without divisors of zero, as well as necessary and sufficient conditions for the imbeddability of a ring in a skew-field. The theory of rings of fractions can also be related to the theory of imbeddings (see Fractions, ring of).

The additive theory of ideals arose in connection with a generalization of the fundamental theorem of arithmetic, which is equivalent to the theorem on representing any ideal of the ring of integers as an intersection of powers of prime ideals, to arbitrary commutative associative rings with the maximum condition (that is, Noetherian rings). The fundamental aim of this theory is to represent any ideal of a ring as the intersection of a finite number of ideals of a certain special form (primary, primal, tertiary, etc.). Here the form of the "special" ideals and the form of the decompositions are chosen so that under certain finiteness conditions the "existence theorems" (that is, any ideal has a decomposition) and the "uniqueness theorems" (with each ideal a certain set of simple ideals is associated that does not depend on the decomposition) will hold. This aim has been achieved for Noetherian rings in the classical Noetherian theory of primary ideals. A generalization of this theorem has been found also for the non-commutative case.

Commutative algebra initially concerned itself with number rings arising in algebraic number theory. Nowadays the theory of commutative rings is a rapidly-developing area on the interface between algebra and algebraic geometry.

Normed, topological, ordered, and certain other rings and algebras with extra structures are often encountered in functional analysis and other areas of mathematics. For further details on rings with additional structures see Normed ring; Topological algebra; Ordered ring.

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Comments[edit]

For non-commutative rings $ R $ acting on a ring $ A $, the bilinearity condition $ \alpha ( ab) = ( \alpha a) b = a( \alpha b) $ is practically incompatible with the module conditions $ ( \alpha + \beta ) a = \alpha a + \beta \alpha $ and $ \alpha ( \beta a ) = ( \alpha \beta ) a $ in that then $ (( \alpha \beta - \beta \alpha ) a ) b = 0 = b (( \alpha \beta - \beta \alpha ) a) $ for all $ a, b \in A $, $ \alpha , \beta \in R $. It is for this reason that the bilinearity condition (2) is not imposed when considering a non-commutative ring of operators $ \Phi $ on a ring $ A $.

The following terminology is used: $ A $ is a ring with operator ring (or ring of operators) $ \Phi $ if conditions (1) hold; $ A $ is a $ \Phi $-algebra, or algebra over $ \Phi $, if (1) and (2) hold.

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