Abstract Algebra Answers

Let a ∈ R, where R is any ring.
(1) Show that if a has a left inverse, then a is not a left 0-divisor.
(2) Show that the converse holds if a ∈ aRa.

What of the following is true for any ring R?
(a) If an is a unit in R, then a is a unit in R.
(b) If a is left-invertible and not a right 0-divisor, then a is a unit in R.
(c) If R is a domain, then R is Dedekind-finite.

Determine is the following True or False: “If ab is a unit, then a, b are units”?

Show that the characteristic of a domain is either 0 or a prime number

Prove that a nonzero ring R is a division ring iff every a ∈ R\{0} is right-invertible

Let (R, +, ×) be a system satisfying all axioms of a ring with identity, except possibly a + b = b + a. Show that a + b = b + a for all a, b ∈ R, so R is indeed a ring

Q.2 (a) How can we differentiate between continuous and discrete random variables. Explain with the help of examples.

3/4(1/2+2/5/1/3)-1/5*3/8

How many concrete blocks of length. 40mm can be cut from a beam of 8.0m in length?

Five years ago, Dylan was half as old as Alyssa was then. Ten years from now, Dylan will be as old as Alyssa was five years ago. How old is Alyssa now?