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Question 1. TT, #, *> is a ring. Given that S=S, +, is a ring with unity e, and (a) The centre of a ring S is defined to be C := {a € S: a. s= sa for every s € S}. Prove that the centre of a ring is a subring of S. What is the centre of a commutative ring S? (b) Show that a subset 7 of a ring S is a subring of S if and only if T is nonempty, T is closed under both the addition and the multiplication of S, and T contains the negative of each of its elements. (c) Prove that SXT is an integral domain if one of S or T is an integral domain and the other contains only a zero element.

Question 1. TT, #, *> is a ring. Given that S=S, +, is a ring with unity e, and (a) The centre of a ring S is defined to be C := {a € S: a. s= sa for every s € S}. Prove that the centre of a ring is a subring of S. What is the centre of a commutative ring S? (b) Show that a subset 7 of a ring S is a subring of S if and only if T is nonempty, T is closed under both the addition and the multiplication of S, and T contains the negative of each of its elements. (c) Prove that SXT is an integral domain if one of S or T is an integral domain and the other contains only a zero element.

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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Question 1. TT, #, *> is a ring. Given that S=S, +, is a ring with unity e, and (a) The centre of a ring S is defined to be C := {a € S: a. s= sa for every s € S}. Prove that the centre of a ring is a subring of S. What is the centre of a commutative ring S? (b) Show that a subset 7 of a ring S is a subring of S if and only if T is nonempty, T is closed under both the addition and the multiplication of S, and T contains the negative of each of its elements. (c) Prove that SXT is an integral domain if one of S or T is an integral domain and the other contains only a zero element.
Transcribed Image Text:Question 1. TT, #, *> is a ring. Given that S=S, +, is a ring with unity e, and (a) The centre of a ring S is defined to be C := {a € S: a. s= sa for every s € S}. Prove that the centre of a ring is a subring of S. What is the centre of a commutative ring S? (b) Show that a subset 7 of a ring S is a subring of S if and only if T is nonempty, T is closed under both the addition and the multiplication of S, and T contains the negative of each of its elements. (c) Prove that SXT is an integral domain if one of S or T is an integral domain and the other contains only a zero element.
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