Let R be a commutative ring which has exactly four ideals {0}, I, J, and R. Among all such rings find a ring which has the smallest number of elements.
Let R be a commutative ring which has exactly four ideals {0}, I, J, and R....
just 10 thank you 9) Let R and S be commutative rings. Show that the cartesian product is a ring with addition and multiplication s') := (r , rrs-s' ) . 10) Let T be a commutative ring containing elements e, f, both 07-such that e+f=h,e=e,f2 = f , and e-f=0T . Show that the ideals R: T e and S T.f are rings but not subrings of T, and that the ring T is isomorphic to the ring 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 1. Prove that Ris a field if and only if the only ideals in Rare (0) and R.
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.
A1. Let M be an R-module and let I, J be ideals in FR (a) Prove that Ann(I +J) -Ann(I) n Ann(J). (b) Prove that Ann(InJ)2 Ann(I) + Ann(J). (c) Give an example where the inclusion in (b) is strict. (d) If R is commutative ald unital and I, J are cornaximal (that is, 1 +J-(1)), prove that Ann(InJ) Ann(I)+Ann(J).
In ring Z36 consider ideals I = (3) and J = (8). (a) Find the order and list all elements of the ideal I. (b)Find the order and list all elements of the ideal J. (c) Is I a maximal ideal? Why? (d) Is I a prime ideal? Why? (e) Is J a maximal ideal? Why? (f) Is J a prime ideal? Why?
Abstract Algebra (1) Let I, J C R be ideals. Show that if I is generated by n elements, and J is generated by m elements, then I +J is generated by no more than nm elements. 1
Part 1 Part 2 7.1.2. Let R be a commutative ring and a, b E R, and define The goal of this problem is to prove that (a, b) is an ideal of R (a) Explain how you know that 0 E (a, b b) What do two random elements of (a, b) look like? Explain why their sum must be in (c) For s E R and z E (a,b), explain why sz E (a, b). 7.2.1. In the...
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].
Solve problem 2 using the priblem 1 . Question is taken from Ring theory dealing with ideals and generating sets for ideals. Problem 1. Suppose that R (R,+ Jis a commutative ring with unity, and suppose F- (a,,. , a } is a finite nonempty subset of R. Modify your proof for Problem 5 above to show that 7n j-1 Problem 2. Consider the set Zo of integer sequences introduced in Homework Problem 6 of Investigation 16. You showed that...