3330_Notes_6o1_fill - Math 3330 Section 6.1 Ideals and...

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Unformatted text preview: Math 3330 Section 6.1 Ideals and Quotient Rings ** Definition – Ideal of a Ring** The subset I of a ring R is called an ideal if it satisfies the following conditions: 1. I is a subring of R 2. For every r R ∈ and for every a I ∈ , ar I ∈ and ra I ∈ Trivial Ideals – If I is an ideal and e I ∈ , then Example: Even integers Alternate Criteria for a subset I of a ring R to be an ideal: Version 1: A. I is nonempty B. For all x and y in I, x + y is in I C. For all x in I, -x is in I D. For all x in I and for all r in R xr and rx are in I. Version 2: A. I is a subgroup of the additive group (R,+) B. For all x in I and for all r in R, rx and xr are in I. Example: Let S be the following subset of the ring of 2-by-2 matrices over the real numbers. : , , a b S a b c c = ∈ ℝ Show S is a subring: Show that the set : 0 x I x = ∈ ℝ is an ideal of S. Show that I is NOT an ideal of the set of all 2-by-2 matrices over the reals. Show that I is NOT an ideal of the set of all 2-by-2 matrices over the reals....
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This note was uploaded on 08/10/2011 for the course MATH 3330 taught by Professor Flagg during the Spring '11 term at University of Houston.

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3330_Notes_6o1_fill - Math 3330 Section 6.1 Ideals and...

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