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For a commutative ring M and ideal A, let N(A)={x in M|there exists a nonnegative integer n such that x^n in A}. Which of following is true for N(A)=A? I. M=Z, A=(2) II. M=Z[x], A=(x^2+2) III. M=Z/27Z, A=(18+27Z) G is a group, a and b are non-unit elements of G, ab=bba. If the subgroup of G generated by a has order 3, what about the order of the subgroup of G generated by b? (a) 3 (b) 5 (c) 7 (d) 9 (e) cannot determined by given information ÇóÖ¸µã£¬Çó½»Á÷£¬Ð»Ð»´ó¼Ò¡£ |
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3Â¥2012-11-07 14:30:12
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I ºÍ II ÒòΪAΪprime ideal. Cannot be determined. Example in Z/105Z, Áía=35, È¡ b=21 »òÕß b=15 ¶¼Âú×ã ab=bba. Á½ÖÖ²»Í¬µÄÈ¡·¨ bµÄorder·Ö±ðΪ5ºÍ7. ÊÂʵÉÏ£¬ ab=bba µÈ¼ÛÓÚ b=a^{-1}b^2a. Òò´Ëb^n=a^-1b^{2n}a. Èç¹ûb^n=e, ÄÇô b^{2n}=e. µÈʽ³ÉÁ¢¡£Òò´ËbµÄorder¿ÉÒÔÊÇÈÎÒâµÄÊý¡£ |
2Â¥2012-11-07 11:01:17









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