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True or False Label each of the following statements as either true or false. 1. Every ideal of a ring R is a subring of R. 2. Every subring of a ring R is an ideal of R. 3. The only ideal of a ring R that contains the unity e is the ring R itself. 4. Any ideal of a ring R is a normal subgroup of the additive group R. 5. The only ideals of the set of real numbers R are the trivial ideals. 6. Every ideal of Z is a principal ideal. 7. For n > 1, the quotient ring of Z by the ideal (n) is Z- 8. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.

True or False Label each of the following statements as either true or false. 1. Every ideal of a ring R is a subring of R. 2. Every subring of a ring R is an ideal of R. 3. The only ideal of a ring R that contains the unity e is the ring R itself. 4. Any ideal of a ring R is a normal subgroup of the additive group R. 5. The only ideals of the set of real numbers R are the trivial ideals. 6. Every ideal of Z is a principal ideal. 7. For n > 1, the quotient ring of Z by the ideal (n) is Z- 8. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.4: Maximal Ideals (optional)
Problem 13E
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True or False Label each of the following statements as either true or false. 1. Every ideal of a ring R is a subring of R. 2. Every subring of a ring R is an ideal of R. 3. The only ideal of a ring R that contains the unity e is the ring R itself. 4. Any ideal of a ring R is a normal subgroup of the additive group R. 5. The only ideals of the set of real numbers R are the trivial ideals. 6. Every ideal of Z is a principal ideal. 7. For n > 1, the quotient ring of Z by the ideal (n) is Z- 8. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.
Transcribed Image Text:True or False Label each of the following statements as either true or false. 1. Every ideal of a ring R is a subring of R. 2. Every subring of a ring R is an ideal of R. 3. The only ideal of a ring R that contains the unity e is the ring R itself. 4. Any ideal of a ring R is a normal subgroup of the additive group R. 5. The only ideals of the set of real numbers R are the trivial ideals. 6. Every ideal of Z is a principal ideal. 7. For n > 1, the quotient ring of Z by the ideal (n) is Z- 8. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.
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