Verify the claim made in Example 10 about the size of RII. Reference: {[a; a:]|aez ]; } , | EXAMPLE 10 Let R = a; and le

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Verify The Claim Made In Example 10 About The Size Of Rii Reference A A Aez Example 10 Let R A And Le 1 (96.7 KiB) Viewed 24 times

Verify the claim made in Example 10 about the size of RII. Reference: {[a; a:]|aez ]; } , | EXAMPLE 10 Let R = a; and let I be the 04 subset of R consisting of matrices with even entries. It is easy to show that I is indeed an ideal of R (Exercise 21). Consider the factor ring RII. The interesting question about this ring is: What is its size? nr We claim RII has 16 elements; in fact, RII +;E{, { 14- An example illustrates the typical situation. Which of the 16 elements 17 8 7 8 is + I? Well, observe that +1= 5 -3 5 -3 16 8 +1= + 1, since an ideal absorbs its own elements. [4 -4] The general case is left to the reader (Exercise 23). ={[2]+1\r:€40 , 11} [ -3) [1] + 5 $ Go