Abstract Algebra | Maximal and prime ideals.

We prove some classic results involving maximal and prime ideals. Specifically we prove the an ideal P is prime iff R/P is an integral domain. Further, we prove that an ideal M is maximal iff R/M is a field.

[wernerprinz]

Sie sind ein ausgezeichneter Mathematiker, der einfach super erklären kann. Genau dieser Fall, wenn R ein Integritätsring oder ein kommutativer Ring mit 1 sei bzw. ein Hauptidealring ist, habe ich das mit den maximalen und primen Idealen nie verstanden, Sie konnten es leicht verständlich und ausführlich erklären, trotz meinen bescheidenen Kenntnissen in Englisch. In Deutschland fand ich hierzu weder in den Fachbüchern noch im deutschen Internet vergeblich eine so einleuchtende Erklärung.

[YitzharVered]

Always great to have some Michael Penn before an abstract Algebra exam! Very helpful for Group theory, lets hope this helps me with rings!

[darrenpeck]

Are we using the correspondence theorem for the maximal ideal and field proof? <a +M> must correspond to I between M and R.

[darrenpeck]

Thank you. Wonderful standard of lecturing.

[madhavestark]

For the prime ideal, shouldn't the ideal be a proper subring of R

[phuocbui]

What is define of <a+M> and <a, M> and why (a +M)(b+M) is in <a+M>?, thank you very much

[asimislam]

Hi Fantastic thanks, please can you tell me (at 11.30) why 1+M must be in the ideal (a, M) ?

You chose b in R to find the inverse of (a+M) but will this mean 1+M has to be in the ideal (a, M) ....please explain, thank you

[darrenpeck]

How do the generators a and M work for the ideal?

[dibeos]

I do not see the difference between the definition of a prime ideal and an ideal (in general). In both cases, whenever ab belongs to an ideal I, either a is in I and b is any element of the ring R, or vice-versa. So what’s the difference between the two definitions?

[natepolidoro]

4:38 It seems like this theorem doesn't use the fact that R is commutative and has an identity, so it is true for all rings R. Am I wrong?

[hybmnzz]

Way too late to say this but the thumbnail has a mistake