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answer number 7,8 plz.Thanks 7- Find all possible value of (-2)/ 8-let R be a commutative...
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
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].
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...
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...
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+...
Help me with the C) please! Only the third one 1. Let R be a commutative ring of characteristic p, a prime a) Prove that (y)y. [3] b) Deduce that the map фр: R-+ R, фр(x)-z", is a ring homomorphism. 1] c) Compute Op in the case R is the ring Zp. [2] d) Prove that фр is injective if R has no zero-divisors. [2] e) Give an example of a commutative ring of characteristic p such that фр is...
10 Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring Ri = S-1R, as in the course. Let D = R/I. Show that the ideal of R1 generated by 1, that is, I R1, is maximal, and RI/I R is isomorphic to the field of fractions of D. (Hint:...
show anc explain in depth every step of the proof requested Let R be a commutative ring with unity Let I be an ideal of R such that if abel then a 6 I or 6 6 I for some nezt Next, let J= {xer | xME I for some mez"} show that I is a Prime ideal of R. (You may assume without proof that I is an ideal in the first place)
= 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 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