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Anonymous No. 16240464

Brainlet here, R cross R is an R-algebra, being a cross product of (trivial) R algebras. It's clearly two dimensional. but this:

https://math.stackexchange.com/questions/2217906/what-are-the-three-non-isomorphic-2-dimensional-algebras-over-mathbbr

shows that all 2d R algebras are either the complex numbers, the set A = {x + jy} where j^2 = 1 or B = {x + ey} where e^2 = 0. Which one of these is RxR then?

Anonymous No. 16240585

>>16240464
A
it's literally written in the response to the question
{x+iy:x,y∈R,i2=1}≅R[x]/(x2−1)≅R[x]/(x−1)×R[x]/(x+1)≅R×R does contain non-trivial idempotents.

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Anonymous No. 16240615

https://en.m.wikipedia.org/wiki/Split-complex_number

The isomorphism is stated right at the top of the article. All you need to understand is that the multiplicative unit in RxR is (1,1) which obviously has more than one "root".