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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove Theorem 5.3: A subset S of the ring R is a subring of R if and only if these conditions are satisfied:
Transcribed Image Text:Prove Theorem 5.3: A subset S of the ring R is a subring of R if and only if these conditions are satisfied:
Ris closed under addition: x e R, y e R = x + y e R Addition in R is associative: (x + y) + z = x+ (y + z) Vx, y, z e R R contains an additive identity 0: x + 0 = 0 + x = x Vx e R R contains an additive inverse: for each x in R, there exists -x in R such that x + (-x) = (-x) + x = 0. Addition in R is commutative: x +y = y+x_Vx, y e R Ris closed under multiplication: x E R, y e R = x · y e R Multiplication in Ris associative: (x · y) · z = x · (y · z) Vx, y, z e R Two distributive laws hold in R: x• (y + z) = x • y + x • z and (x+ y) · z = x · z +y•z Vx, y, z E R
Transcribed Image Text:Ris closed under addition: x e R, y e R = x + y e R Addition in R is associative: (x + y) + z = x+ (y + z) Vx, y, z e R R contains an additive identity 0: x + 0 = 0 + x = x Vx e R R contains an additive inverse: for each x in R, there exists -x in R such that x + (-x) = (-x) + x = 0. Addition in R is commutative: x +y = y+x_Vx, y e R Ris closed under multiplication: x E R, y e R = x · y e R Multiplication in Ris associative: (x · y) · z = x · (y · z) Vx, y, z e R Two distributive laws hold in R: x• (y + z) = x • y + x • z and (x+ y) · z = x · z +y•z Vx, y, z E R
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