3. Consider the ring R- Zz[x] and the ideal Ixx+1>, (a) Is I a prime ideal?...
12. NEZ True] [False] A maximal ideal is prime. [True] [False] The ring Q[x]/<r? + 10x + 5) is a field [True] [False] If R is an integral domain and I c R is an ideal, then R/I is an integral domain as well [True] [False] The map : M2(Q) - Q defined by °(A) = det(A) is a ring homomorphism. [True] [False] If I, J are distinct ideals of a ring R then the quotient rings R/T and R/T...
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:...
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. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].
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?
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...
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....
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....
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. (a) Let p > 2 be prime. Describe all groups of order p. (b) Give two examples of non-isomorphic groups of order p?, explaining clearly why they are non-isomorphic. (c) For which pairs of primes p >q is there a unique group of order pg? (a) Classify all groups of order 4907 explaining clearly all steps of your argument.