On Tue, Jan 5, 2021 at 10:56 PM Dan Asimov <asimov@msri.org> wrote:
I would find the definition more compelling if it didn't have the condition "when R is nonempty, then".
I think you misunderstand; the definition doesn't contain the phrase you mention above. The definition is just:
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,
It is then a theorem that:
when R is nonempty, it has an additive unit given by 0 = g.
This definition is almost equivalent to the standard definition, the only difference being that this definition allows the empty set (with the obvious definitions of addition and multiplication) to be considered a ring.. Andy
(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.
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