Notes:Rings and u-rings

Notes

• A ring homomorphism requiring:
  1. [ilmath]f(x+y)=f(x)+f(y)[/ilmath] and
  2. [ilmath]f(xy)=f(x)f(y)[/ilmath]

  For u-rings [ilmath]A[/ilmath] and [ilmath]B[/ilmath] and [ilmath]f:A\rightarrow B[/ilmath] need not be a u-ring homomorphism! There's a distinction.

• We must add:
  3. [ilmath]f(1)=1[/ilmath]

  For it to be a u-ring morphism.

• Proof:
  • Consider [ilmath]f:\mathbb{R}\rightarrow\mathbb{R}\times\mathbb{R} [/ilmath] by [ilmath]f:x\mapsto(x,0)[/ilmath]. This satisfies 1 and 2, but not 3.

Questions

Are u-modules and modules distinct in this way?