Abstract Algebra Answers

I need help in finding the average in 3 3/4, 2 1/4, 2, 4 1/2, 5/6, and 2/3. My problem come in when I add the bottom roll of fractions together. then what do I do?

the cost of item is increased by 2times, after the increasing, the item costs 2 times expensive than the first, what percent is the amount increased in the first time, if the second time the amount is increased by 25%

And the answer is 60 !!
so how to solve this ? I tried to solve it and i got 12.5 ! so please with a explanation write the steps down for me

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