Abstract Algebra Answers

Let A be an algebra over a field k such that every element of A is algebraic over k. Let B be a subalgebra of A, and b ∈ B. Show that b is a unit in B iff it is a unit in A.

Let A be an algebra over a field k such that every element of A is algebraic over k. Show that a nonzero element of A is a unit iff it is not a 0-divisor.

Let A be an algebraic algebra over a field k. Show that A is Dedekind-finite.

Show that any left noetherian ring R is Dedekind-finite.

Show that the left regular module R is hopfian iff R is Dedekind - finite.

Show that any noetherian module M is hopfian.

sqrt(x^2)=|x| how can i proof it by real analysis

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