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2. Show that P = {(a,-a, 0,0): a € Z} with addition and multiplication defined by (a,-a, 0,0) + (b, -b, 0,0) = (a + b, -a- b,0,0) (a, -a, 0, 0) (b, -b, 0,0) = (ab, -ab, 0,0) is a subring of M in #1. Is P commutative or non-commutative? Why?

2. Show that P = {(a,-a, 0,0): a € Z} with addition and multiplication defined by (a,-a, 0,0) + (b, -b, 0,0) = (a + b, -a- b,0,0) (a, -a, 0, 0) (b, -b, 0,0) = (ab, -ab, 0,0) is a subring of M in #1. Is P commutative or non-commutative? Why?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.1: Sets And Geometry
Problem 10E: Consider sets A, B, and C from Exercise 9. Find: a AB b BC c ABAC A=1,2,3,4. B=2,4,6,8, and...
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Answer no. 2 only.

1. Consider the set M = {(a, b, c, 0): a, b, c E Q} with addition and multiplication defined by (a, b, c,0) + (e, f, g,0) = (a + e,b+f,c + g,0) (a, b, c, 0) (e, f, g,0) = (ae + bg, af, ce, cf) for all (a, b, c, 0), (e, f, g, h) E M. Show that M is a ring under the defined operations addition and multiplication. 2. Show that P = {(a, -a, 0,0): a € Z} with addition and multiplication defined by (a, -a, 0, 0) + (b,-b, 0,0) = (a + b,-a - b,0,0) (a, -a, 0, 0) (b, -b, 0,0) = (ab, -ab, 0,0) is a subring of M in #1. Is P commutative or non-commutative? Why?
Transcribed Image Text:1. Consider the set M = {(a, b, c, 0): a, b, c E Q} with addition and multiplication defined by (a, b, c,0) + (e, f, g,0) = (a + e,b+f,c + g,0) (a, b, c, 0) (e, f, g,0) = (ae + bg, af, ce, cf) for all (a, b, c, 0), (e, f, g, h) E M. Show that M is a ring under the defined operations addition and multiplication. 2. Show that P = {(a, -a, 0,0): a € Z} with addition and multiplication defined by (a, -a, 0, 0) + (b,-b, 0,0) = (a + b,-a - b,0,0) (a, -a, 0, 0) (b, -b, 0,0) = (ab, -ab, 0,0) is a subring of M in #1. Is P commutative or non-commutative? Why?
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