I would find the definition more compelling if it didn't have the condition "when R is nonempty, then". (E.g., the definition of a power set holds for all sets, including all the ones without elements.) —Dan
On Tuesday/5January/2021, at 7:34 PM, Mike Stay <metaweta@gmail.com> wrote:
On Tue, Jan 5, 2021 at 5:30 PM Allan Wechsler <acwacw@gmail.com <mailto:acwacw@gmail.com>> wrote:
There aren't any rings with no elements, are there?
Not as traditionally defined, but we can use a definition derived from the idea of a group with no elements to define a notion of a ring with no elements. https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html <https://golem.ph.utexas.edu/category/2020/08/the_group_with_no_elements.html>
If we say a ring is a set R equipped with an associative commutative binary operation +: R x R -> R, a binary operation -: R x R -> R such that ∀g, h. g + (h - g) = h ∀g, h. (g + h) - g = h and a binary operation *: R x R -> R such that ∀g, h, j. g * (h + j) = g * h + g * j and ∀g, h, j. (h + j) * g = h * g + j * g,
then when R is nonempty, it must have an additive unit given by 0 = g - g for any g. But the definition works just fine for the empty R as well.