Lecture 23 was difficult but it was the last lecture in E-222 on group theory. The remaining lectures will be on rings which are also ( abelian ) groups but with an additional operation ( multiplication ). Purpose of the first lecture on rings was mainly to establish some definitions and show analogies between group theory and ring theory. I expect a lot of stuff on ideals in the forthcoming lectures.
A ring is an abelian group with an additional operation * such that:
- * is closed.
- * is associative
- 1 * a = a.
This makes the ring a monoid with respect to *.
( Gross defined it slightly different in the lecture ).
Gross showed an interesting way to construct rings from abelian rings. The endomorphisms of an abelian group End(A) form a ring if an element f_a in the ring is constructed from an element a in A by taking the map f(1) = a.
Some furter topics which were introduced:
( Ideal )
( Quotient ring )
( Unit )
( Examples of groups of units )
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