These are chat archives for non/algebra

Mar 2018
Srepfler Srdan
Mar 16 2018 00:36
what do you mean @jeremyrsmith by "to be an additive or multiplicative group in the absence of the other (within a ring)”?
Mar 16 2018 00:41
just that they’re structurally indistinguishable from one another, unless they’re both part of a ring structure
if you have a ring structure, then you have two group structures and you know which is additive and which is multiplicative by the nature of multiplication distributing over addition (at least)
if all you have is a group structure, there is no structural way to say whether it’s additive or multiplicative, without “peeking"
(as far as I can determine)
Srepfler Srdan
Mar 16 2018 00:43
I think in abstract algebra the terms additiona and multiplication are mostly inherited from historical baggage of dealing with numbers
a Zero and One are both a Neutral element of the group
and Negate and Invert are the Reciprocal of the value relative to the Neutral
the fact that you have two operations which follow these rules
that defines the Ring
No grups, no ring
both addition and multiplications are a function from two operands to one
Srepfler Srdan
Mar 16 2018 00:53
So I imagine what the guys want to say is here's a framework on how to reason if computations which are chained have a result in the end since it will tell you if you have holes in your domain and arrival set
and then the category theory guys kind of expanded that
Denis Rosset
Mar 16 2018 01:22
There are a few things between "commutative ring" and "field", such as "unique factorization domain" and "euclidean ring".
Wikipedia is pretty complete on that