Q1) If (R, +,.) is integral domain, then the characteristic of R is either zero or prime. ==== Q2) Let (P(X),A,n) be a ring, where X {1, 2, 3}. Answer the following. (i) Find the zero element 0 and the identity element 1 in the ring P(X). (ii) Find the additive inverse of the element {1,3} in P(X). (iii) Find the characteristic of P(X). (iv) Find all the zero divisors. Q3) Any ring R is without zero divisor if and only if the cancellation laws for multiplication =ca where a 0 implies b = c. are satisfied, that is for a, b, c ER, ab = ac and ba =x, then Char(R) = 2. Q4) Let R be a ring with identity R. If for every x R it holds x²

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 20E
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Q1) If (R, +,.) is integral domain, then the characteristic of R is either zero or prime.
====
Q2) Let (P(X),A,n) be a ring, where X {1, 2, 3}. Answer the following. (i) Find the zero
element 0 and the identity element 1 in the ring P(X). (ii) Find the additive inverse of the
element {1,3} in P(X). (iii) Find the characteristic of P(X). (iv) Find all the zero divisors.
Q3) Any ring R is without zero divisor if and only if the cancellation laws for multiplication
=ca where a 0 implies b = c.
are satisfied, that is for a, b, c ER, ab = ac and ba
=x, then Char(R) = 2.
Q4) Let R be a ring with identity R. If for every x R it holds x²
Transcribed Image Text:Q1) If (R, +,.) is integral domain, then the characteristic of R is either zero or prime. ==== Q2) Let (P(X),A,n) be a ring, where X {1, 2, 3}. Answer the following. (i) Find the zero element 0 and the identity element 1 in the ring P(X). (ii) Find the additive inverse of the element {1,3} in P(X). (iii) Find the characteristic of P(X). (iv) Find all the zero divisors. Q3) Any ring R is without zero divisor if and only if the cancellation laws for multiplication =ca where a 0 implies b = c. are satisfied, that is for a, b, c ER, ab = ac and ba =x, then Char(R) = 2. Q4) Let R be a ring with identity R. If for every x R it holds x²
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