Fichas
Rings: Ideals
Fichas del conjunto (13)
- 1A subring I of R is a left ideal ifFor all a in I, and r in R, ra is in I
- 2A subring I of R is a left ideal ifFor all a in I, and r in R, ar is in I
- 3An ideal (\(I \triangleleft R\))isA subring that is both a left and right ideal
- 4\(I \triangleleft R\) is proper if\(I\subsetneqR\)
- 5\(I \triangleleft R\) is trivial ifI = {0}
- 6Prove that an ideal is proper iff \(1 \notin I \)For all r in R, if 1 is in I then r1 = r is in I so I = R. The converse if self-obvious.
- 7Prove that in a division ring, the only proper ideal is trivial.Suppose we have an ideal in a division with a nonzero element a. Take any element b in our division ring. Then \(a^{-1}b\) is in the division ring as well, and \(aa^{-1}b=b\) is in the ideal. Therefore, it is not a proper ideal.
- 8Let S be a nonempty subset of a ring R. Then the ideal generated S is defined to bethe smallest ideal in R containing S, which would be the intersection of all such ideals.
- 9Characterize the ideal generated by S using collections of all finite sums\(\{\sum_{i=1}^n r_{i1}s_ir_{i2} |r_{i1},r_{i2} \in R, s_i \in S \}\)
- 10All ideals containing S mustContain the ideal generated by S
- 11If the ideal generated by S is commutative, then one can simply characterize it as\(\{\sum_{i=1}^n r_{i}1s_i|r_{i} \in R, s_i \in S \}\)
- 12The ideal generated by a single element a is calledA principal ideal
- 13An integral domain in which every ideal is a principal ideal is calleda principal ideal domain
Información
- Descripción
- Undergrad Math Fichas sobre Rings: Ideals, creado por Vincent Tembo el 26/07/2016.
- Creado por
- Vincent Tembo
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