View source for Near-ring

One of the generalizations of the concept of an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]). A near-ring is a [[Ringoid|ringoid]] over a group, i.e. a [[Universal algebra|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

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

Examples of near-rings are the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661402.png" /> of all mappings of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661403.png" /> into itself which commute with the action of a given semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661404.png" /> of endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661405.png" />. The group operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661406.png" /> are defined pointwise and multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661407.png" /> is composition of mappings. A near-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661408.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661409.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614010.png" />) be the variety of near-rings defined by the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614012.png" />). Every near-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614013.png" /> can be decomposed into the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614014.png" /> of sub-near-rings, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614017.png" />. A cyclic right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614018.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614019.png" /> is called primitive of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614022.png" /> is simple; primitive of type 1 if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614023.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614024.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614025.png" />; and primitive of type 2 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614026.png" /> is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614027.png" />-module. A near-ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614028.png" /> is called primitive of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614030.png" />) if there is a faithful simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614031.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614032.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614033.png" />. In this case there is a dense imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614034.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614035.png" /> for some semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614036.png" /> of endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614037.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614038.png" />-primitive near-rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614039.png" /> with an identity element and with the minimum condition for right ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614040.png" />, the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614041.png" /> holds (an analogue of the Wedderburn–Artin theorem). For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614042.png" />, the Jacobson radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614044.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614045.png" /> can be introduced as the intersection of the annihilators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614046.png" />-primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614047.png" />-modules. The radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614048.png" /> is defined as the intersection of the maximal right module ideals. All four radicals are different, and

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

For near-rings an analogue of Ore's theorem on near-rings of fractions [[#References|[4]]] holds.

A distributively-generated near-ring is a near-ring whose additive group is generated by elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614050.png" /> such that

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614053.png" /> in the near-ring. All distributively-generated near-rings generate the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614054.png" />. For finite distributively-generated near-rings the notions of 1- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614055.png" />-primitivity coincide; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614056.png" />-primitive distributively-generated near-rings have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614057.png" /> for some group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614058.png" />. In a distributively-generated near-ring with the identity

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

Every near-ring from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614060.png" /> without nilpotent elements is a subdirect product of near-rings without divisors of zero [[#References|[4]]]. A near-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614061.png" /> 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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614062.png" /> does not contain ideals with zero multiplication; and c) any annihilator of any minimal ideal is maximal [[#References|[1]]].

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

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