symmetric monoidal (∞,1)-category of spectra
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
and also
Canceling $x$ on the left and $y$ on the right, we have $x+y=y+x$.
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.
A near-ring is a set $R$ equipped with
A group structure $(R,+,0)$,
A monoid structure $(R,\cdot,1)$,
such that for any $x,y,z\in R$ we have $(x+y)\cdot z = (x\cdot z) + (y\cdot z)$, and for any $x\in R$ we have $0\cdot x = 0$.
If $(R,+,0)$ is only a monoid or semigroup, we say instead that $R$ 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 $(R,+,0)$ is weakened to a loop algebraic structure (i.e. left and right additive inverses no longer coincide and the associativity of addition is dropped).
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.
For loop near rings see
Loop near rings appear in topology:
Last revised on June 10, 2021 at 11:56:51. See the history of this page for a list of all contributions to it.