Abstract Algebra Answers

Let I be a left ideal in a ring R such that, for some integer n ≥ 2, an = 0 for all a ∈ I. Show that an−1Ran−1 = 0 for all a ∈ I.

For any ring R and any ordinal α, define Nα(R) as follows. For α = 1, N1(R) is a nil subideal of Nil*R which contains all nilpotent one-sided ideals of R. If α is the successor of an ordinal β, define
Nα(R) = {r ∈ R : r + Nβ(R) ∈ N1 (R/Nβ(R))}. If α is a limit ordinal, define Nα(R) = (Union over β<α) Nβ(R).
Show that Nil*R = Nα(R) for any ordinal α with Card α > Card R.

Give an example of a (necessarily noncommutative) ring R in which N1(R) is strictly contained Nil*R.

Let N1(R) be the sum of all nilpotent ideals in a ring R.
Show that the hypothesis and conclusion in (2) both apply if the ideals in R satisfy DCC.

Let N1(R) be the sum of all nilpotent ideals in a ring R. If N1(R) is nilpotent, show that N1(R) = Nil*R.

Let N1(R) be the sum of all nilpotent ideals in a ring R. Show that N1(R) is a nil subideal of Nil*R which contains all nilpotent one-sided ideals of R.

Let I ⊆ R be a right ideal containing no nonzero nilpotent right ideals of R. (For instance, I may be any right ideal in a semiprime ring.) Show that the following are equivalent:
(1) IR is an artinian module;
(2) IR is a finitely generated semisimple module. In this case, show that
(3) I = eR for an idempotent e ∈ I.

the radius of the field in the form of a sector is 21 meter .thecost of constructing of wall around hte field is 1875 rs. at the rate of 25 per meter if is cost 10 per m^2 to till the field ,what will be the cost of tilling the whole filed?

Q 1)
Let I and J be (left or right or two-sided) ideals of a ring R. We define their product IJ to be a set {x1y1 + . . . + xnyn | xi ∈ I, yj ∈ J}.
Show that the set IJ is again an (left, right or two-sided) ideal. Moreover, show that (IJ)K = I(JK) for any ideal I, J, K of R

Q 2)
The composition ′◦′ satisfies the following two axioms
1. (associativity) If f: A → B, g: B → C and h: C → D, then h ◦ (g ◦ f) = (h ◦ g) ◦ f.
2. (identity) For every object A ∈ ob(C) there exists a map 1A : A → A called the identity map for A, such that for every morphism f : A → B we have 1B ◦ f = f ◦ 1A = f
Show using the axioms that the identity map is unique for every object.

Q3)
Check that R[G] satisfies the ring axioms.(R[G] is
a group ring of G over R).

Assume that y varies directly with x. If y = 910 when x =2, y=34 when x =7.