Label each of the following statements as either true or false.
The only ideal of a ring
Whether the statement, ‘The only ideal of a ring
Answer to Problem 1TFE
Solution:
The statement, ‘The only ideal of a ring
Explanation of Solution
Consider the statement, “The only ideal of a ring
Definition of maximal ideal: Let
Therefore, the only ideal of a ring
Hence the statement, ‘The only ideal of a ring
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Chapter 6 Solutions
Elements Of Modern Algebra
- True or false Label each of the following statements as either true or false. 3. The only ideal of a ring that contains the unity is the ring itself.arrow_forwardTrue or False Label each of the following statements as either true or false. Every ideal of a ring is a subring of.arrow_forwardExercises If and are two ideals of the ring , prove that is an ideal of .arrow_forward
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- Label each of the following statements as either true or false. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forwardLet I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.arrow_forward
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forwardLabel each of the following statements as either true or false. Every subring of a ring R is an idea of R.arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,