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Ring

A ring is a non-empty set typeset structure equipped with two binary operations 1 , commonly called multiplication and addition, and written as a product typeset structure and as a sum typeset structure, satisfying the following conditions:

  1. typeset structure is a commutative group, that is
    1. addition typeset structure is associative: for all typeset structure, typeset structure
    2. addition typeset structure is commutative:  for all typeset structure, typeset structure
    3. there exists an additive identity, i.e. a unit element with respect to +, usually called zero of the ring typeset structure and denoted typeset structure such that for all typeset structure, typeset structure
    4. for every typeset structure there exists an additive inverse typeset structure, denoted typeset structure, such that typeset structure
  2. typeset structure is a grupoid. It is customary to denote the product typeset structure only by typeset structure.
  3. Multiplication is left and right distributive with respect to the multiplication: for all typeset structure we have
    1. typeset structure
    2. typeset structure

If in addition

  1. typeset structure is a semigroup, that is the multiplication typeset structure is associative: for all typeset structure, typeset structure,

then typeset structure 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 typeset structure and typeset structure be two rings. Let typeset structure be the set of mappings of typeset structure into typeset structure. Define for typeset structure

(f ◊ g) (x) = f(x) ⊕ g(x)       and       (f * g) (x) = f(x) O· g(x)

for all typeset structure. Then typeset structure  is a ring under the above binary operations typeset structure and typeset structure.  For instance, its zero (additive unit) is the constant map whose value is the zero of typeset structure.  If  typeset structure is associative then so is also typeset structure.

If

  1. typeset structure is a loop, that is the multiplication possesses an identity element (unit) typeset structure, i.e. for all typeset structure, typeset structure,  

the ring typeset structure is termed a ring with identity, or also a unit ring. 2

If in the above Example 1 the ring  typeset structure possesses a multiplicative identity, say typeset structure,  then typeset structure is a ring with identity whose multiplicative identity is the constant map whose value is typeset structure.

Even integers form a ring without identity.

If the multiplication is commutative, that is

  1. for all typeset structure, typeset structure,

the ring is called a commutative ring. Many texts additionally require that a commutative ring is a ring with identity.

Example 2: Let typeset structure be an additive abelian group. Let typeset structure be the set of group-homomorphisms of typeset structure into itself. Define an addition in typeset structure to be the addition of mappings, that is every typeset structure, typeset structure for all typeset structure, and their multiplication to be the composition of mappings, typeset structure for all typeset structure. Then typeset structure is a ring, whose multiplicative identity is the identity map (typeset structure for all typeset structure).  In general, typeset structure is not multiplicative.

If  typeset structure is a ring with identity typeset structure with typeset structure such that

  1.   typeset structure is a group, that is
    1. ring multiplication is associative
    2. for every typeset structure there exists a multiplicative inverse typeset structure, usually denoted typeset structure, i.e. typeset structure

then typeset structure is termed a division ring or a skew field.  

Let typeset structure be a ring with identity typeset structure. The set typeset structure of elements of typeset structure which have both a right and left inverse is a multiplicative group. The set is called the group of units of typeset structure or the group of invertible elements of typeset structure.  It is often denoted by typeset structure.

A division ring is a ring typeset structure with identity typeset structure such that typeset structure.

A skew field whose multiplication is commutative is called a field  3 .

A subset typeset structure of a ring typeset structure is called a subring of typeset structure  if typeset structure itself is a ring with respect to the induced operations of addition and multiplication. In other words, if typeset structure is an additive subgroup of typeset structure and typeset structure is a subsemigroup of typeset structure. 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 typeset structure is the subset of typeset structure consisting of all elements typeset structure such that typeset structure for all typeset structure. The center of ring is a subring.

Let typeset structure be a ring with identity. A left ideal typeset structure in typeset structure is a subset of typeset structure which is a subgroup of the additive group of typeset structure, and such that typeset structure is a subset of typeset structure (since typeset structure contains identity, typeset structure). To define a right ideal we require that typeset structure. A two sided-ideal  or shortly an ideal of typeset structure is a subset which is both a left and a right ideal of typeset structure.

If typeset structure is a ring with identity and typeset structure, then the set typeset structure is an left ideal of typeset structure termed left principal ideal. Similarly we define a right principal ideal or a principal ideal.   

The subset typeset structure and typeset structure are ideals in every ring typeset structure. If typeset structure is a division ring or a field, then these are its only ideals. A ring typeset structure with no two-sided ideals different from typeset structure and typeset structure is called simple.

A commutative ring with identity typeset structure such that its every ideal is principal is called a principal ideal ring (PIR).

Notes

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 typeset structure.  Here the successive powers of the added element typeset structure eventually loops around to get something already existing: typeset structure is a “new” number, while typeset structure is already available in typeset structure. 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 typeset structure or typeset structure.

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.

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