Euclidian rings extend rings with operations for division and
remainder, with the Euclidian definition:
plus(mul(div(s, t), t), mod(s, t)) === s,
f(mod(s, t)) in [0, abs(t)) for some appropriate
embedding into real numbers.
Fields are commutative rings in which all non-zero elements have multiplicative inverses.
Groups are monoids that additionally have inverses
Monoids are semigroups with a neutral element (or zero)
Ordered rings are rings with ordering relation in which
addition, multiplication, and ordering are consistent:
leq(s, t) ==> leq(plus(s, a), plus(t, a)) and
leq(zero, s) & leq(zero, t) ==> leq(zero, mul(s, t)).
A Pseudo-ring is a structure with the same operations as a ring, but no guarantee that multiplication satisfies the ring axioms
Rings are structures with both addition and multiplication
Rings that also have a division operation (though possibly not satisfying the standard axioms)
Ring that can also convert ring elements back to integers.
Rings that also possess an ordering relation