If B and C are A -algebras with ring morphisms f : A &#x2192;<!-- → --> B and

Lillianna Andersen 2022-07-15 Answered
If B and C are A-algebras with ring morphisms f : A B and g : A C, and D = B A C is an A-algebra with morphism a f ( a ) g ( a ), then u f = v g, where u : B D is u ( b ) = b 1.The map v : C D is not defined in the text, but my guess is it's v ( c ) = 1 c.I don't understand why the diagram is commutative though. That would imply f ( a ) 1 = 1 g ( a ) for all a A. Is that true, or is v something else?Added: On second thought, does this follow since f ( a ) 1 = a ( 1 1 ) and 1 g ( a ) = a ( 1 1 ) where ⋅ is the A-module structure on D?
A f B g # u C v D
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Answers (1)

diamondogsaz
Answered 2022-07-16 Author has 12 answers
In their definition for algebras, they let f : A B be a ring homomorphism; if a A, b B, define a product a b = f ( a ) b.
So we have
u f ( a ) = f ( a ) 1 C = f ( a ) 1 B 1 C = a 1 B 1 C = a ( 1 B 1 C )
and
v g ( a ) = 1 B g ( a ) = 1 B 1 C g ( a ) = 1 B 1 C a = ( 1 B 1 C ) a
but D = B R C is a commutative ring with identity 1 B 1 C so you can conclude that a ( 1 B 1 C ) = ( 1 B 1 C ) a and hence u f = v g as desired.
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