Ideals and Quotient Rings -- Abstract Algebra Examples 19

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[terryendicott]

This was an example of an ideal lecture.

[schweinmachtbree]

Two mistakes in the penultimate example: the obvious one is at 19:59 which should be _s_ = _a_ _u_ ^(-1) _y_ ϵ ( _a_ ). the other one is at 17:52 where we can only cancel _a_ if it is non-zero, so we split into two cases: either _a_ ≠ 0 and we proceed as Michael does, or _a_ = 0 in which case ( _b_ ) = (0) ⇒ _b_ ϵ (0) = {0} so _b_ = 0 too, and so _a_ and _b_ are equal up to multiplication by some unit _u_ ; any unit works.

Also we have actually done the second example before in the group theory section of the course; there we phrased things in terms of cyclic subgroups as <m> ∩ <n> = <l> and <m> + <n> = <d>, but cyclic subgroups in the group ( *Z*, +) are the same as principal ideals in the ring ( *Z*, +, ×) (when considered as sets): <k> = { n·k | n ϵ *Z* } = { n×k | n ϵ *Z* } = (k).

[alegal]

17:52 mistake: a could be zero, in that case you can't left cancel by a.

[ko-prometheus]

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Hello,
What can abstract algebra give in the analysis of a function, a group of functions?
Can I analyze 50 functions relative to each other!! and get the output from the sum we outputs of all 50 functions??