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Ring Theory

Rings of Unity, Monoid, Momomorphism and Invertible Elements

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

Ring Homomorphisms

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'.

Rings and Subrings Functions

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

Euler Tour : Dominoes

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

The Heat Equation on a Metal Ring

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

Quotient Ring

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

Describe the Rings

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.