6. Let R be a non-trivial ring with unity. Suppose x E R satisfies x2 =...
(5) Suppose R is a commutative ring with unity, and r e R. Let A(r) {s E R : rs-0). Prove that A(r) is an ideal of R.
Let R be a ring with identity 1. Suppose that 08 a € R satisfies a? = a. Show that for each TER, there exists a positive integer n such that [(1 – a)ral" = 0. What is the smallest possible value of n that works for all r ER?
Let R be a ring with unity 1, and let u ∈ R be a unit. Let ne be a positive integer so that nu = 0. Show that na = 0 for all a ∈ 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.
let R be a ring with unity (180) and let UE U(R). Suppose that I is an ideal of R sot. UE Io prove that I =R
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
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
Thanks 6. Let R be a ring and a € R. Prove that (i) {x E R | ax = 0} is a right ideal of R (ii) {Y E R | ya=0} is a left ideal of R (iii) if L is a left ideal of R, then {z E R za = 0 Vae L} is a two-sided ideal of R NB: first show that each set in 6.(i), (ii), (iii) above is a subring T ool of...
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].
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...