One of the generalizations of the concept of an associative ring (cf. Associative rings and algebras). A near-ring is a ringoid over a group, i.e. a universal algebra in which an associative multiplication and addition exist; a near-ring is a (not necessarily Abelian) group with respect to addition, and the right distributive property
must hold too. A near-ring is also an example of a multi-operator group.
Examples of near-rings are the set of all mappings of a group into itself which commute with the action of a given semi-group of endomorphisms of . The group operations in are defined pointwise and multiplication in is composition of mappings. A near-ring is an analogue of a ring of matrices. The notions of a sub-near-ring, of an ideal and of a right module over a near-ring are introduced in the usual manner.
Let () be the variety of near-rings defined by the identity (). Every near-ring can be decomposed into the sum of sub-near-rings, where , and . A cyclic right -module is called primitive of type if is simple; primitive of type 1 if either or for any ; and primitive of type 2 if is a simple -module. A near-ring is called primitive of type () if there is a faithful simple -module of type . In this case there is a dense imbedding of into for some semi-group of endomorphisms of . For -primitive near-rings with an identity element and with the minimum condition for right ideals in , the equality holds (an analogue of the Wedderburn–Artin theorem). For every , the Jacobson radical of type can be introduced as the intersection of the annihilators of -primitive -modules. The radical is defined as the intersection of the maximal right module ideals. All four radicals are different, and
For near-rings an analogue of Ore's theorem on near-rings of fractions  holds.
A distributively-generated near-ring is a near-ring whose additive group is generated by elements such that
for all and in the near-ring. All distributively-generated near-rings generate the variety . For finite distributively-generated near-rings the notions of 1- and -primitivity coincide; -primitive distributively-generated near-rings have the form for some group . In a distributively-generated near-ring with the identity
Every near-ring from without nilpotent elements is a subdirect product of near-rings without divisors of zero . A near-algebra can be decomposed into a direct sum of simple near-rings if and only if: a) it satisfies the minimum condition for principal ideals; b) does not contain ideals with zero multiplication; and c) any annihilator of any minimal ideal is maximal .
For near-rings one can prove results similar to those on the structure of regular rings  and on near-rings of fractions . Near-rings have applications in the study of permutation groups, block-schemes and projective geometry .
|||H.E. Bell, "A commutativity theorem for near-rings" Canad. Math. Bull. , 20 : 1 (1977) pp. 25–28|
|||H.E. Heatherly, "Regular near-rings" J. Indian Math. Soc. , 38 (1974) pp. 345–354|
|||S. Ligh, "The structure of certain classes of rings and near rings" J. London Math. Soc. , 12 : 1 (1975) pp. 27–31|
|||G. Pilz, "Near-rings" , North-Holland (1983)|
|||A. Oswald, "On near-rings of quotients" Proc. Edinburgh Math. Soc. , 22 : 2 (1979) pp. 77–86|
|||S.V. Polin, "Generalizations of rings" , Rings , 1 , Novosibirsk (1973) pp. 41–45 (In Russian)|
|||J.D.P. Meldrum, "Near-rings and their links with groups" , Pitman (1985)|
Near-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Near-ring&oldid=15999