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1 Local nearrings

A set R with two binary operations + and \cdot is called a (left) nearring if the following statements hold:

1. (R,+)=R^+ is a (not necessarily abelian) group with neutral element 0;

2. (R,\cdot) is a semigroup;

3. x(y+z)=xy+xz for all x, y, z\in R.

If R is a nearring, then the group R^+ is called the additive group of R. If in addition 0\cdot x=0, then the nearring R is called zero-symmetric, and if the semigroup (R,\cdot) is a monoid, i.e. it has an identity element i, then R is a nearring with identity i. In the latter case the group R^* of all invertible elements of the monoid (R,\cdot) is called the multiplicative group of R.

The concepts of a subnearring and a nearring homomorphism are defined by the same way as for rings. In particular, if \lambda is a nearring homomorphism of (R,+, \cdot), then its kernel Ker \lambda is a subnearring of (R,+, \cdot) whose additive subgroup is normal in R^+.

A subnearring I of (R,+, \cdot) is an ideal of (R,+, \cdot) if I = Ker \lambda for some \lambda.

It can simply be verified that I is an ideal of R if and only if its additive group I^+ is a normal subgroup of R^+ and for any elements r, s\in R and a\in I the inclusions ra\in I and (r + a)s − rs\in I hold. Main results accumulated for local nearrings can be found in the surveys [Sys08] and [RR25].

A nearring R with identity is said to be local if the set L=R\setminus R^* of all non-invertible elements of R is a subgroup of R^+.

It is clear that if L is an ideal of R, then the factor nearring R/L is a nearfield. For example, every local ring R is a zero-symmetric local nearring whose subgroup L coincides with the Jacobson radical of R. Reference: [Max68].

1.1 The local nearrings library

1.1-1 AdditiveGroupsOfLibraryOfLNRsOfOrder

Returns: a list

The argument is n. The output is a list of additive groups of local nearrings in the library of this package of order n.

gap> List(AdditiveGroupsOfLibraryOfLNRsOfOrder(81),IdGroup);
[ [ 81, 1 ], [ 81, 2 ], [ 81, 3 ], [ 81, 5 ], [ 81, 6 ], [ 81, 11 ], 
  [ 81, 12 ], [ 81, 13 ], [ 81, 15 ] ]

1.1-2 LibraryOfLNRsOnGroup

Returns: a list

The argument is a group G. The output is a list of catalogue entries for the local nearrings in the library of this package whose additive group is isomorphic to G.

The local nearrings are sorted by their multiplicative groups.

gap> G:=SmallGroup(81,2);
<pc group of size 81 with 4 generators>
gap> LibraryOfLNRsOnGroup(G);
[ "AllLocalNearRings(81,2,54,3)", "AllLocalNearRings(81,2,54,6)", 
  "AllLocalNearRings(81,2,54,9)", "AllLocalNearRings(81,2,54,10)", 
  "AllLocalNearRings(81,2,54,11)", "AllLocalNearRings(81,2,54,15)", 
  "AllLocalNearRings(81,2,72,14)", "AllLocalNearRings(81,2,72,19)", 
  "AllLocalNearRings(81,2,72,24)", "AllLocalNearRings(81,2,72,26)" ]

1.1-3 LocalNearRing

Returns: a nearring

The arguments are k, l, m, n, w. The output is the w-th local nearring from the library of this package whose additive group has IdGroup value [k,l] and whose multiplicative group has IdGroup value [m,n]. No validation of the arguments is performed.

gap> L:=LocalNearRing(81,12,54,8,3);
ExplicitMultiplicationNearRing ( <pc group of size 81 with 
4 generators> , multiplication )

1.1-4 AllLocalNearRings

Returns: a list

The arguments are k, l, m, n. The output is the list of all local nearrings from the library of this package whose additive group has IdGroup value [k,l] and whose multiplicative group has IdGroup value [m,n]. No validation of the arguments is performed.

gap> L:=AllLocalNearRings(81,12,54,8);;
gap> Size(L);
30

1.1-5 NumberLocalNearRings

Returns: a number

The arguments are k, l, m, n. The output is the number of local nearrings in the library of this package whose additive group has IdGroup value [k,l] and whose multiplicative group has IdGroup value [m,n]. No validation of the arguments is performed.

gap> NumberLocalNearRings(81,15,54,8);
10

1.1-6 IsAdditiveGroupOfLibraryOfLNRs

Returns: a boolean

The argument is a group G. The output is true if the library of this package contains a local nearring whose additive group is isomorphic to G, and false otherwise.

gap> G:=SmallGroup(25,2);
<pc group of size 25 with 2 generators>
gap> IsAdditiveGroupOfLibraryOfLNRs(G);
true
gap> IsAdditiveGroupOfLibraryOfLNRs(SmallGroup(81,14));
false

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