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4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a multiplication as follows: (A, z) + (µ, w) := (A+µ,z +w) and (λ, 2) . (μ, w) :(μ, λυ + μ). Show that: (i) R with these two operations is a commutative ring; (ii) R, {(0,0)}, and I:= { (0, 2) | z € R} (0, z) are the only ideals in R; and (iii) I = Nil(R).

4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a multiplication as follows: (A, z) + (µ, w) := (A+µ,z +w) and (λ, 2) . (μ, w) :(μ, λυ + μ). Show that: (i) R with these two operations is a commutative ring; (ii) R, {(0,0)}, and I:= { (0, 2) | z € R} (0, z) are the only ideals in R; and (iii) I = Nil(R).

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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4) Define on the cartesian product of sets {(A, 2) | A, 2 € R } R:= R x R an addition and a multiplication as follows: (A, z) + (µ, w) := (X+µ, z + w) and ( λ, 2) . ( μ w) :(Αμ, λω + μ2) . Show that: (i) R with these two operations is a commutative ring; (ii) R, {(0,0)}, and I := { (0, 2)| z ER} are the only ideals in R; and (0, 2) ) z € (iii) I = Nil(R).
Transcribed Image Text:4) Define on the cartesian product of sets {(A, 2) | A, 2 € R } R:= R x R an addition and a multiplication as follows: (A, z) + (µ, w) := (X+µ, z + w) and ( λ, 2) . ( μ w) :(Αμ, λω + μ2) . Show that: (i) R with these two operations is a commutative ring; (ii) R, {(0,0)}, and I := { (0, 2)| z ER} are the only ideals in R; and (0, 2) ) z € (iii) I = Nil(R).
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