### Polynomial Rings, Finite Dimensional Vector Space

Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.

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Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.

(1) Given a ring R, an element e is called an idempotent if e^2 = e. (i) Let R1 and R2 be two commutative rings with unity. Consider R = R1 x R2. Find two non-zero idempotents e1,e2 E R such that 1= e1 + e2 and e1e2 = 0. (Be careful: what is 1 in this R? What is 0?) (ii) On the other hand, suppose R is any commutative ring wit

Note: Z is integer numbers C is set containment Here is the problem Let I be an ideal in a ring R. Define [ R : I ] = { r in R such that xr in R for all x in R } 1) Show that [ R : I ] is an ideal of R that contains I 2) If R is assumed to have a unity, what can you say about [ R : I ] ? 3) Find [ 2Z :

Modern Algebra Division Ring

Let R commutative ring with unity, and S a sub monoid of the multiplicative monoid of R. In RxS define (a,b) ~ (b,t) if Эu є S э u(at-bs)= 0. Show that ~ is an equivalence relation in RxS. Denote the equivalence class of (a,s) as a/s and the quotient set consisting of these classes as RS-¹. Show that RS-¹ be

A cylindrical drill with radius 2 is used to bore a hole throught the center of a sphere of radius 7. Find the volume of the ring shaped solid that remains using polar coordinates.

Let phi be a homomorphism of a ring R with unity onto a nonzero ring R'. Let u be a unit in R. Show that phi (u) is a unit in R'.

----------------- Let R1 and R2 be integral domains with quotient fields F1 and F2 respectively. If phi: R1 -> R2 is a ring isomorphism, show that phi extends to an isomorphism phi hat : F1 -> F2. Here extends means that phi hat(a) = phi (a) for all a in R1 (Hint: under the givien assumptions, there is really only one way to

Show that if D is an integral domain, then {n.1 | n is interger} is a subdomain of D contained in every subdomain of D.

A) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains. b) Express the volume in part (a) in terms of the height h of the ring. Notice that the volume depends only on h, not on r1 and r2.

Please see the attached file for the fully formatted problems. 1. Ler R be a ring, and , prove, using axioms for a ring, the following ? The identity element of R s unique ? That -r is the unique element of R such tht (-r)+r = 0. (hint, for part 1, suppose that 1 and 1' ate two identities of R, show that 1-1' must be

Let p be a prime. Show that in the ring Z-p (set of integers modulo p) we have (a+b)^p = a^p+b^p for all a, b in Z-p. The following hint was given: observe that the usual binomial expansion for (a+b)^n is valid in a commutative ring.

2. A domino is a 2x1 rectangular piece of wood. On each half of the domino is a number, denoted by dots. In the figure, we show all C(5,2) = 10 dominoes we can make where the numbers on the dominoes are all pairs of values chosen from {1,2,3,4,5} (we do not include dominoes where the two numbers are the same). Notice that we hav

I need to prove the following: Show that if U is the collection of all units in a ring <R, +, *> with unity, then <U,*> is a group. A reminder was given to make sure to show that U is closed under multiplication.

Let u(x,t) describe the temperature of a thin metal ring with circumference 2pi. For convenience, let's orient the ring so that x spans the interval |-pi, pi|. Suppose that the ring has some internal heating that is angle-dependent, so that u(x, t) satisfies the inhomogeneous heat equation u_t = ku_zz + f(x), where k is t

Problem: Prove the Second Isomorphism Theorem: If A is an ideal of R and S is a subring of R, then S+A is a subring, A, and (S intersecting A) are ideals of S+A and S, respectively, and (S+A)/A isomorphic to A/(S intersecting A).

Problem: Let R be a commutative ring. Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal. ALSO is every prime ideal of Z(integers) maximal? Why?

Please see the attached PDF file. I would prefer a solution in PDF format. Thanks!

Please see the attached file for the fully formatted problem. Let gcd(m, n) = 1. We know that Zmn = Zm × Zn as an additive group. Is there a ring isomorphism Zmn = Zm × Zn? Same question for the complex numbers C with C = R × R as an additive group.

Please see the attached file for the fully formatted problems. Describe the ring obtained from Z12 by adjoining the element 1/2 (the inverse of 2).

Please see the attached file for the fully formatted problems. Prove that Z5/(x2 + x + 1) is a field. How many elements are there in this field? Can you also represent it as Z5[x]/(x2-a) where a is some element of Z5?

Please see the attached file for the fully formatted problem. Describe the rings: Z[x]/(x2 − 3, 2x + 4), Z[i]/(2 + i) where i2 = −1.