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].
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+...
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.
(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.
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....
Please answer all parts. Thank you! 20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
answer number 7,8 plz.Thanks 7- Find all possible value of (-2)/ 8-let R be a commutative ring, and let I be an ideal in R. If P is a prime ideal of 1, prove that p is an ideal of R
Let R be Commutative ring with 1 and let N and M be two R-modules Prove that NM MBN Let R be Commutative ring with 1 and let N and M be two R-modules Prove that NM MBN
= 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...