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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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n0661401.png" /></td> </tr></table>
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$$x(y+z)=xy+xz$$
  
 
must hold too. A near-ring is also an example of a [[Multi-operator group|multi-operator group]].
 
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.
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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 <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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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614049.png" /></td> </tr></table>
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$$J_0(A)\subseteq J_{1/2}(A)\subseteq J_1(A)\subseteq J_2(A).$$
  
 
It turns out that these radicals posses many properties of the [[Jacobson radical|Jacobson radical]] of an associative ring (cf. [[#References|[4]]]).
 
It turns out that these radicals posses many properties of the [[Jacobson radical|Jacobson radical]] of an associative ring (cf. [[#References|[4]]]).
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For near-rings an analogue of Ore's theorem on near-rings of fractions [[#References|[4]]] holds.
 
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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A distributively-generated near-ring is a near-ring whose additive group is generated by elements $x$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614051.png" /></td> </tr></table>
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$$(y+z)x=yx+zx$$
  
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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066140/n06614059.png" /></td> </tr></table>
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$$(xy-yx)^{n(x,y)}=xy-yx,\quad n(x,y)>1,$$
  
 
multiplication is commutative (cf. [[#References|[3]]], [[#References|[4]]]).
 
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]]].
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Every near-ring from $N_0$ without nilpotent elements is a subdirect product of near-rings without divisors of zero [[#References|[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 [[#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]]].
 
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]]].

Latest revision as of 11:23, 27 October 2014

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)
How to Cite This Entry:
Near-ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Near-ring&oldid=34096
This article was adapted from an original article by V.A. Artamonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article