Let I be an ideals of a ring R and S C R: (i) Show that InS is an ideal of S. Is it also an ideal of R? (ii) Either prove or give counterexample that every ideal of S is of the type InS for some ideal I of R. (iii) Show that if I n S homomorphism. {OR} then 7(S) = S, where T : R → R/I is the projection %3D
Let I be an ideals of a ring R and S C R: (i) Show that InS is an ideal of S. Is it also an ideal of R? (ii) Either prove or give counterexample that every ideal of S is of the type InS for some ideal I of R. (iii) Show that if I n S homomorphism. {OR} then 7(S) = S, where T : R → R/I is the projection %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

Transcribed Image Text:Let I be an ideals of a ring R and SE R:
(i) Show that InS is an ideal of S. Is it also an ideal of R?
(ii) Either prove or give counterexample that every ideal of S is of the type INS for some
ideal I of R.
{OR} then 7(S) = S, where T : R → R/I is the projection
(iii) Show that if InS =
homomorphism.
Expert Solution

Step 1
I is an ideal of a ring R and S is subring of R.
So if
and
Hence,
Similarly
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