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A ring is a non-empty set equipped with two binary operations 1 , commonly called multiplication and addition, and written as a product and as a sum , satisfying the following conditions:
If in addition
then is called an associative ring. Rings in which the multiplication is non-associative are considered only in special situations. Practically a ring is almost always defined as an associative ring.
Example 1: Let and be two rings. Let be the set of mappings of into . Define for
for all . Then is a ring under the above binary operations and . For instance, its zero (additive unit) is the constant map whose value is the zero of . If is associative then so is also .
If
the ring is termed a ring with identity, or also a unit ring. 2
If in the above Example 1 the ring possesses a multiplicative identity, say , then is a ring with identity whose multiplicative identity is the constant map whose value is .
Even integers form a ring without identity.
If the multiplication is commutative, that is
the ring is called a commutative ring. Many texts additionally require that a commutative ring is a ring with identity.
Example 2: Let be an additive abelian group. Let be the set of group-homomorphisms of into itself. Define an addition in to be the addition of mappings, that is every , for all , and their multiplication to be the composition of mappings, for all . Then is a ring, whose multiplicative identity is the identity map ( for all ). In general, is not multiplicative.
If is a ring with identity with such that
then is termed a division ring or a skew field.
Let be a ring with identity . The set of elements of which have both a right and left inverse is a multiplicative group. The set is called the group of units of or the group of invertible elements of . It is often denoted by .
A division ring is a ring with identity such that .
A skew field whose multiplication is commutative is called a field 3 .
A subset of a ring is called a subring of if itself is a ring with respect to the induced operations of addition and multiplication. In other words, if is an additive subgroup of and is a subsemigroup of . In a similar manner the notion of a subfield is defined as a subset of a field which itself is a field with respect to the induced operations of addition and multiplication.
The center of a ring is the subset of consisting of all elements such that for all . The center of ring is a subring.
Let be a ring with identity. A left ideal in is a subset of which is a subgroup of the additive group of , and such that is a subset of (since contains identity, ). To define a right ideal we require that . A two sided-ideal or shortly an ideal of is a subset which is both a left and a right ideal of .
If is a ring with identity and , then the set is an left ideal of termed left principal ideal. Similarly we define a right principal ideal or a principal ideal.
The subset and are ideals in every ring . If is a division ring or a field, then these are its only ideals. A ring with no two-sided ideals different from and is called simple.
A commutative ring with identity such that its every ideal is principal is called a principal ideal ring (PIR).
1 | The word ring has its origin in the German word Zahlring (=number ring). The term Zahlring was introduced by D.Hilbert (DieTheorie der algebraischen Zahlen, Bericht der Deutschen Mathematischen Vereinigung 1894/95, Chapter IX) for objects called number rings in number theory. As an example we can take . Here the successive powers of the added element eventually loops around to get something already existing: is a “new” number, while is already available in . Probably the first who used this term in this abstract meaning was Adolf Fraenkel in 1914 (Über die Teiler der Null und die Zerlegung von Ringen. J. reine angew. Math. 145 (1915), 139-176; Dissertation Marburg, 43 pp. 1914). The forerunner of this notion was R.Dedekind’s term order (Ordnung) in the algebraic number theory (Über die Anzahl der Idealklassen in den verschiedenen Ordnungen eines endlichen Körpers, Festschrift zur Säkularfeier von Gauß‘ Geburtstag, Braunschweig 1877). |
2 | For instance, Bourbaki calls rings without identity a pseudo-rings. There also exists a jocular term rng , i.e. a ring without the multiplicative identity often denoted by or . |
3 | The term field developed from the German Körper (Körper=solid body) introduced by R.Dedekind into algebraic number theory in his edition of Dirichlet’s Vorlesungen über Zahlentheorie (2th ed. Braunschweig 1871, p. 424). According to F.Klein (Vorlesungen über die Entwicklung der Mathematik im 19.Jahrhundert, p.320) the term Körper should remind the word Körperschaft (=companionship or body) due to a common feature of algebraic numbers that they can be together rationally combined. Remember the original English terms domain of rationality or realm of rationality. These, however, go back to the German Rationalitätsbereich used by Kronecker. The first general definition of a Körper was given by H.Weber (Die allgemeinen Grundlagen der Galoisschen Gleichungstheorie, Math. Ann. 43, 1893). The notion was then deeply studied by E.Steinitz in the period 1908-1910 (Algebraische Theorie der Körper, J. für Math. 137 (1910) 167-309; new edition with an appendix on Galois theory by R. Baer und H. Hasse published by W. de Gruyter & Co. Berlin 1930). |
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Štefan Porubský: Ring.