Please help!
Let phi: R rightarrow S be a surjective homomorphism from the ring R to the ring S. Prove that if T is a subring of R, then the set W= {b S such that b = phi (c) for some c T } is a subring of S. [Be specific of how and where you use that T is a subring of R.]
Solution
Let a, b in W. Then a= phi(r) for some r in T and b = phi(t) for some t in T. Now to show a+b is in W, notice that a+b = phi(r)+phi(t)=phi(r+t). Since T is a subring and r and t are in T, then r+t is in T too. So a+b is in W. To show a*b is in W, notice that since T is a subring, r*t is in W. So a*b = phi(r)phi(t)=phi(rt). SInce rt is in T, a*b is inW. To show that -a is in W, notice that since T is a subring, if r is in T, so is -r. So, phi(-r) = -phi(r) = -a. Hence -a is in W. Finally, since T is a subring, 0 is in T. So 0 = phi(0), so 0 is in W too. By the subgring criterion, W is a subring. Please rate :).
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Please help! Let phi- R rightarrow S be a surjective homomorphism from.docx
1. Please help!
Let phi: R rightarrow S be a surjective homomorphism from the ring R to the ring S. Prove that if
T is a subring of R, then the set W= {b S such that b = phi (c) for some c T } is a subring of S.
[Be specific of how and where you use that T is a subring of R.]
Solution
Let a, b in W. Then a= phi(r) for some r in T and b = phi(t) for some t in T. Now to show a+b is
in W, notice that a+b = phi(r)+phi(t)=phi(r+t). Since T is a subring and r and t are in T, then r+t
is in T too. So a+b is in W. To show a*b is in W, notice that since T is a subring, r*t is in W. So
a*b = phi(r)phi(t)=phi(rt). SInce rt is in T, a*b is inW. To show that -a is in W, notice that since
T is a subring, if r is in T, so is -r. So, phi(-r) = -phi(r) = -a. Hence -a is in W. Finally, since T is a
subring, 0 is in T. So 0 = phi(0), so 0 is in W too. By the subgring criterion, W is a subring.
Please rate :).
![Please help!
Let phi: R rightarrow S be a surjective homomorphism from the ring R to the ring S. Prove that if
T is a subring of R, then the set W= {b S such that b = phi (c) for some c T } is a subring of S.
[Be specific of how and where you use that T is a subring of R.]
Solution
Let a, b in W. Then a= phi(r) for some r in T and b = phi(t) for some t in T. Now to show a+b is
in W, notice that a+b = phi(r)+phi(t)=phi(r+t). Since T is a subring and r and t are in T, then r+t
is in T too. So a+b is in W. To show a*b is in W, notice that since T is a subring, r*t is in W. So
a*b = phi(r)phi(t)=phi(rt). SInce rt is in T, a*b is inW. To show that -a is in W, notice that since
T is a subring, if r is in T, so is -r. So, phi(-r) = -phi(r) = -a. Hence -a is in W. Finally, since T is a
subring, 0 is in T. So 0 = phi(0), so 0 is in W too. By the subgring criterion, W is a subring.
Please rate :).](https://image.slidesharecdn.com/pleasehelpletphi-rrightarrowsbeasurjectivehomomorphismfrom-230129071206-2e9abd04/85/Please-help-Let-phi-R-rightarrow-S-be-a-surjective-homomorphism-from-docx-1-320.jpg)
