I have the solutions to this problem. Please provide me with 2-3 examples Prove Theorem 5.3: A subset S of the ring R is a subring of R if and only if these conditions aresatisfied: Ris closed under addition: x e R, y e R = x + y e RAddition in R is associative: (x + y) + z = x+ (y + z) Vx, y, z e RR contains an additive identity 0: x + 0 = 0 + x = x Vx e RR contains an additive inverse: for each x in R, there exists -x in Rsuch that x + (-x) = (-x) + x = 0.Addition in R is commutative: x +y = y+x_Vx, y e RRis closed under multiplication: x E R, y e R = x · y e RMultiplication in Ris associative: (x · y) · z = x · (y · z) Vx, y, z e RTwo distributive laws hold in R:x• (y + z) = x • y + x • z and (x+ y) · z = x · z +y•zVx, y, z E R

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Description: I have the solutions to this problem. Please provide me with 2-3 examples Prove Theorem 5.3: A subset S of the ring R is a subring of R if and only if these conditions aresatisfied: Ris closed under addition: x e R, y e R = x + y e RAddition in R is

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 10E: Exercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these...

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