Near-ring

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

It turns out that these radicals posses many properties of the Jacobson radical of an associative ring (cf. [4]).

For near-rings an analogue of Ore's theorem on near-rings of fractions [4] 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

multiplication is commutative (cf. [3], [4]).

Every near-ring from without nilpotent elements is a subdirect product of near-rings without divisors of zero [4]. 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 [1].

For near-rings one can prove results similar to those on the structure of regular rings [2] and on near-rings of fractions [5]. Near-rings have applications in the study of permutation groups, block-schemes and projective geometry [4].

References

[1]	H.E. Bell, "A commutativity theorem for near-rings" Canad. Math. Bull. , 20 : 1 (1977) pp. 25–28
[2]	H.E. Heatherly, "Regular near-rings" J. Indian Math. Soc. , 38 (1974) pp. 345–354
[3]	S. Ligh, "The structure of certain classes of rings and near rings" J. London Math. Soc. , 12 : 1 (1975) pp. 27–31
[4]	G. Pilz, "Near-rings" , North-Holland (1983)
[5]	A. Oswald, "On near-rings of quotients" Proc. Edinburgh Math. Soc. , 22 : 2 (1979) pp. 77–86
[6]	S.V. Polin, "Generalizations of rings" , Rings , 1 , Novosibirsk (1973) pp. 41–45 (In Russian)
[7]	J.D.P. Meldrum, "Near-rings and their links with groups" , Pitman (1985)