proof that every subring of a cyclic ring is an ideal
The following is a proof that every subring of a cyclic ring is an ideal.
Let be a cyclic ring and be a subring of . Then and are both cyclic rings. Let be a generator (http://planetmath.org/Generator) of the additive group of and be a generator of the additive group of . Then . Thus, there exists with .
Since is a ring, . Thus, there exists with . Since , it follows that is an ideal of . ∎
- 1 Buck, Warren. http://planetmath.org/?op=getobj&from=papers&id=336Cyclic Rings. Charleston, IL: Eastern Illinois University, 2004.
- 2 Maurer, I. Gy. and Vincze, J. “Despre Inele Ciclece.” Studia Universitatis Babeş-Bolyai. Series Mathematica-Physica, vol. 9 #1. Cluj, Romania: Universitatea Babeş-Bolyai, 1964, pp. 25-27.
|Title||proof that every subring of a cyclic ring is an ideal|
|Date of creation||2013-03-22 13:30:52|
|Last modified on||2013-03-22 13:30:52|
|Last modified by||Wkbj79 (1863)|