4. An element a in a ring R is called nilpotent if there exists a non-negative...
> 0. Show that the 7. An element of a ring is nilpotent when " = 0 for some nilpotent elements of a commutative ring R constitute an ideal of R.
Fact: If d > 2 is an integer, then there exists a prime q such that q divides d. (1) Let e and f be positive integers. Prove that if ged(e, f) = 1, then god(e?,f) = 1. (2) Let m be a positive integer. Prove that if m is rational, then m is an integer.
(a) An element in a ring R is nilptent in there exists n e Z such that " = 0. The nilradical N of R is the set of its nilpotent elements. Find the nilradical of Z12. (b) Describe the ring Za[i]/(3+ i). (c) For which integers n does x2 + 2x + 1 divide x1 +52 + 2x2 + 3x + 15 in Z/nZ[:)?
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...
2. Given a ring (R, +,-) with IR and OR to be the identity w.r.t . and +. Define (-1) = -1 and inductively for k > 1 that kr:= (k – 1)r +1R (-k)R:= (-k+1)R+(-1)R. Define the map f:Z + R from the ring of integers to R to be f(n) = nR E R. Prove that f is a ring homomorphism. (Hint: use induction somewhere).
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...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
Let R be a commutative ring with no nonzero zero divisor and elements r1,r2,.. . ,Tn where n is a positive integer and n 2. In this problem you will sketch a proof that R is a field (a) We first show that R has a multiplicative identity. Sinee the additive identity of R is, there is a nonzero a E R. Consider the elements ari, ar2, ..., arn. These are distinct. To see O. Since R conelude that0, which...
10. A natural number n is called attainable if there exists non-negative integers a and b such that n - 5a + 8b. Otherwise, n is called unattainable. Construct an 9 x 6 matrix whose rows are indexed by the integers between 0 and 8 and whose columns are indexed by the integers between 0 and 5 whose (x, y)-th entry equals 5x + 8y for any 0 < r < 8 and (a) Mark down all the attainable numbers...
Definition A: Let R be ring and r e R. Then r is called a zero-divisor in Rifr+0r and there exists SER with s # OR and rs = OR. Exercise 1. Let R be a ring with identity and f € R[2]. Prove or give a counter example: (a) If f is a zero-divisor in R[x], then lead(f) is a zero-divisor in R. (b) If lead(f) is a zero- divisor in R[x], then f is a zero-divisor in R[2]....