Let K be the factor ring of the ting R=(a + bila.be Z) by the ideal...
QUESTION 4 (a) Let RS be a ring homomorphism with I an ideal of R and J an ideal of S. Define 0(I) = {$(1) I ET) and o-'(J) = {ve R(y) € J} and check as to whether or not (i) °(1) is an ideal of S (6) (ii) o-'() is an ideal of R (6) (Hint: I, J are two-sided ideals and in both cases of (i) and (ii) above, first check the subring conditions) (b) Given a...
= Let R be a ring (not necessarily commutative) and let I be a two-sided ideal in R. Let 0 : R + R/I denote the natural projection homomorphism, and write ř = º(r) = r +I. (a) Show that the function Ø : Mn(R) + Mn(R/I) M = (mij) Ø(M)= M is a surjective ring homomorphism with ker ý = Mn(I). (b) Use Homework 11, Problem 2, to argue that M2(2Z) is a maximal ideal in M2(Z). (c) Show...
Let R be a commutative ring with unity 1 and let I be a minimal ideal in R i.e. a nonzero ideal which does not properly contain another non-zero ideal. Show that either the product of two elements in I is always zero or there is an element in I that serves as unity in the ring I. Show also that in the latter case I is a field.
IBMISSION Аа Let R be a ring and let I be an ideal of R. Recall the canonical map T: RR/I given by *(r) = r + I is a ring homomorphism from R to R/I that is onto R/I, but not one-to-one. Let S be a subring of R where I SS. Part 1: Explain why +(S) is a subring of R/I. (Hint: Don't make this difficult. Can you use the corollary in the previous video?) INSTRUCTIONS Make a...
5. Let I be an ideal in a ring R. Prove that the natural ring homomorphism T: RRI has kernel equal to I.
3. Consider the ring R- Zz[x] and the ideal Ixx+1>, (a) Is I a prime ideal? Is I a maximal ideal? (b) Find all the multiplicative units of R/I (a+ bx cx2 a, b, c E 2. Is the group of units cyclic? If so, give a generator. If not, determine to what commor group it is isomorphic?
2. Let R be a commutative ring with unity 1, and let a be a unit in R Let / be an ideal in R that contains the element a. Prove that / cannot be a proper ideal of R. 3. Let R be a commutative ring with unity 1 of order 30, and let be a prime ideal of R. Prove that is a maximal ideal of R
74. Let R be a commutative ring with identity such that not every ideal is a principal ideal principal ideal. (b) If I is the ideal of part (a), show that R/I is a principal ideal ring
74. Let R be a commutative ring with identity such that not every ideal is a principal ideal principal ideal. (b) If I is the ideal of part (a), show that R/I is a principal ideal ring
subring of the polynomial ring R{z] (i Show that R is a (ii) Let k be a fixed positive integer and Rrk be the set of all polynomials of degree less than or subring of Ra (iii) Find the quotient q(x) and remainder r(x) of the polynomial P\(x) 2x in Z11] equal to k. Is Rr]k a T52r43 -5 when divided by P2(x) = iv) List all the polynomials of degree 3 in Z2[r].
subring of the polynomial ring R{z]...
QUESTION C.
(a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,...,zn]. 2. (b) State what it means for a ring R to be Noetherian. (c) State Hilbert's basis theorem. Give a proof of Hilbert's basis theorem using the fact if k is a field the polynomial ring kli,..., In] is Noetherian. 1S
(a) Let k be a field and let n be a positive integer. Define...