nLab
near-ring

Near-rings

Idea

A near-ring is a sort of ring in which addition need not be commutative. However, if we simply remove the commutativity of addition from the usual definition of a ring, then nothing changes, because we can prove commutativity from the other axioms: we have

(x+y)(1+1)=x(1+1)+y(1+1)=x+x+y+y (x+y)(1+1) = x(1+1) + y(1+1) = x + x + y + y

and also

(x+y)(1+1)=(x+y)1+(x+y)1=x+y+x+y. (x+y)(1+1) = (x+y)1 + (x+y)1 = x + y + x + y.

Canceling xx on the left and yy on the right, we have x+y=y+xx+y=y+x.

Thus, in order for the notion of near-ring to be different from that of a ring, we need to relax the distributivity law as well; we impose it only on one side.

Definition

A near-ring is a set RR equipped with

  1. A group structure (R,+,0)(R,+,0),

  2. A monoid structure (R,,1)(R,\cdot,1),

  3. such that for any x,y,zRx,y,z\in R we have (x+y)z=(xz)+(yz)(x+y)\cdot z = (x\cdot z) + (y\cdot z), and for any xRx\in R we have 0x=00\cdot x = 0.

If (R,+,0)(R,+,0) is only a monoid or semigroup, we say instead that RR is a near-rig or a near-semiring. (Of course, now it is possible to have distributivity on both sides without making addition commutative, since addition need not always cancel.)

There is also a weaker notion, of loop near ring, where in the axioms above the group structure on (R,+,0)(R,+,0) is weakened to a loop algebraic structure (i.e. the associativity of addition is dropped).

Internalization

Of course, near-rings can be defined internally to any cartesian monoidal category. More generally, they can be defined internally to a braided duoidal category.

References

For loop near rings see

  • D. Ramakotaiah, C. Santhakumari. On loop near-rings. Bull. Austral. Math. Soc. 19(3):417{435, 1978.

Loop near rings appear in topology:

  • Damir Franetič, Petar Pavešić, Loop near-rings and unique decompositions of H-spaces, arxiv/1511.06168
category: algebra

Last revised on May 21, 2016 at 14:00:01. See the history of this page for a list of all contributions to it.