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Math / Contemporary Abstract Algebra 8 / Chapter 14 / Problem 3E

Contemporary Abstract Algebra | 8th Edition | ISBN: 9781133599708 | Authors: Joseph Gallian

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Verify that the set I in Example 5 is an ideal and that if

Chapter 14, Problem 3E

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QUESTION:

Problem 3E

Verify that the set I in Example 5 is an ideal and that if J is any ideal of R that contains a1, a2, . . . , an, then I ⊆J. (Hence, ka1, a2, . . . , anl is the smallest ideal of R that contains a1, a2, . . . , an.)

Reference:

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QUESTION:

Problem 3E

Verify that the set I in Example 5 is an ideal and that if J is any ideal of R that contains a1, a2, . . . , an, then I ⊆J. (Hence, ka1, a2, . . . , anl is the smallest ideal of R that contains a1, a2, . . . , an.)

Reference:

ANSWER:

Step 1 of 2

Check the set is an ideal and it is the smallest ideal containing .

Clearly, is not empty. For any two elements and in ,

As is a ring, the subtraction of two elements and belongs to . So, , and . Thus, is an ideal of

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