Find a ring R with a proper ideal I and an element b such that b is not a unit in R but (b + I) is a unit in R/I.
Find a ring R with a proper ideal I and an element b such that b is not a unit in R but (b + I) is a unit in R/I.
thanks Let I be a proper ideal of a commutative ring R with 1. Prove that I is a maximal 3. (10 ideal of R if and only if for every a e R\I, I+(a) : {i+ ar i e I,rE R} = R. Let I be a proper ideal of a commutative ring R with 1. Prove that I is a maximal 3. (10 ideal of R if and only if for every a e R\I, I+(a) : {i+...
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
66. Let R be a commutative ring with identity. An ideal I of R is irreducible if it cannot be expressed as the intersection of two ideals of R neither of which is contained in the other. the following. (a) If P is a prime ideal then P is irreducible. (b) If z is a non-zero element of R, then there is an ideal I, maximal with respect to the property that r gI, and I is irreducible. (c) If...
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.
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
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?
10. If I is a nonzero ideal in a Dedekind domain R, then R/I is an Artinian ring. 10. If I is a nonzero ideal in a Dedekind domain R, then R/I is an Artinian ring.
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 commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].
5. Let I be an ideal in a ring R. Prove that the natural ring homomorphism T: RRI has kernel equal to I.