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

$$x(y+z)=xy+xz$$

must hold too. A near-ring is also an example of a multi-operator group.

Examples of near-rings are the set $M_S(\Gamma)$ of all mappings of a group $\Gamma$ into itself which commute with the action of a given semi-group $S$ of endomorphisms of $\Gamma$. The group operations in $M_S(\Gamma)$ are defined pointwise and multiplication in $M_S(\Gamma)$ is composition of mappings. A near-ring $M_S(\Gamma)$ 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 $N_0$ ($N_c$) be the variety of near-rings defined by the identity $0x=0$ ($0x=x$). Every near-ring $A$ can be decomposed into the sum $A=A_0+A_c$ of sub-near-rings, where $A_0\in N_0$, $A_c\in N_c$ and $A_0\cap A_c=0$. A cyclic right $A$-module $M$ is called primitive of type $0$ if $M$ is simple; primitive of type 1 if either $xA=0$ or $xA=M$ for any $x\in M$; and primitive of type 2 if $M$ is a simple $A_0$-module. A near-ring $A$ is called primitive of type $\nu$ ($\nu=0,1,2$) if there is a faithful simple $A$-module $\Gamma$ of type $\nu$. In this case there is a dense imbedding of $A$ into $M_S(\Gamma)$ for some semi-group $S$ of endomorphisms of $\Gamma$. For $2$-primitive near-rings $A$ with an identity element and with the minimum condition for right ideals in $A_0$, the equality $A=M_S(\Gamma)$ holds (an analogue of the Wedderburn–Artin theorem). For every $\nu=0,1,2$, the Jacobson radical $J_\nu(A)$ of type $\nu$ can be introduced as the intersection of the annihilators of $\nu$-primitive $A$-modules. The radical $J_{1/2}(A)$ is defined as the intersection of the maximal right module ideals. All four radicals are different, and

$$J_0(A)\subseteq J_{1/2}(A)\subseteq J_1(A)\subseteq J_2(A).$$

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 $x$ such that

$$(y+z)x=yx+zx$$

for all $y$ and $z$ in the near-ring. All distributively-generated near-rings generate the variety $N_0$. For finite distributively-generated near-rings the notions of $1$- and $2$-primitivity coincide; $1$-primitive distributively-generated near-rings have the form $M_0(\Gamma)$ for some group $\Gamma$. In a distributively-generated near-ring with the identity

$$(xy-yx)^{n(x,y)}=xy-yx,\quad n(x,y)>1,$$

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

Every near-ring from $N_0$ without nilpotent elements is a subdirect product of near-rings without divisors of zero [4]. A near-algebra $A$ 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) $A$ 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)