Algerbraic Structures: Subrings, Zero-divisors, Rings, Groups and Fields

A4. Which of the following subsets of Q are subrings of Q? Give reasons for your answers.... A5. Find (i) the number of zero-divisors in Z100 and (ii) the number of units in Z100 ,giving reasons for your answers. Please see the attached file for the fully formatted problems. keywords: zero, divisors

Algerbraic Structures: Groups and Orders of Elements

Define what is meant by the order of an element in a group. Show that if x is an element of finite order in a group and the order of x is the same as the order of x^2 then the order of x is odd. Find the order of each of the following elements in the given groups, giving reasons for your answers.... Please see the attached f ...continues

Groups rings and fields

Please help with the following questions: B7 Define what it means for a group to be cyclic.....

Algerbraic Structures: Units and Zero-divisor in a Commutative Ring with Identity

B8. Define what is meant by (a) a unit, (b) a zero-divisor in a commutative ring with identity. Show that an element in a communtative ring with identity (where 1 = 0) cannot be both a unit and a zero-divisor. Find an element in Z...Z which is neither a unit nor a zero-divisor. Show that... is a unit in .... Find a unit . ...continues

Symmetric groups

symmetric groups: G = Sn. (i) Let g1, g2 belong G be two disjoint cycles, and let g = g1g2. Prove that o(g) = lcm { o( g1), o(g2)}, where lcm stands for the least common multiple. (ii) Let g= g1g2 ... gr belong G, where g1,g2, ... gr are disjoint cycles. Prove that o(g) = lcm {o(g1), o(g2), ... o(gr)}. Can you ...continues

Isomorphism

This week lecture is taught about Isomorphism, automorphism and Inner automorphism, but I don't understand what they are. Can you give some simple examples?

Ring Theory/Largest two-sided ideal

Let I be a right ideal of a ring R and let A = {r in R: (R/I)r = 0}. Prove that A is the largest two-sided ideal of R contained in I.

Ring Theory/Nil radical

Let R be a ring with the property that every element is either nilpotent or invertible. If a, b, c are in R with a and b nilpotent, show that ac, ca, and a + b are nilpotent. For the latter, first observe that a + b cannot equal 1. Conclude that Nil (R) is the set of all nilpotent elements of R. (nil radical Nil (R) is def ...continues

Prove that if G is a group of order n and F is any field then GLn(F) contains

Prove that if G is a group of order n and F is any field then GLn(F) contains a subgroup isomorphic to G.

Group of order 9

This is the question: Consider small groups. (i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3 (ii) List all groups of order at most 10 (up to isomorphism)