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Random Talk on Random Thoughts

Saw a New Type of Ring

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This Thursday evening, I saw a local ring in a commutative ring with unity for the first time. At the first glance, I didn’t know how to make use of it’s definition — a ring that has only one maximal ideal — to answer a question. If I was given a local ring $R$, then should I first assume that there’s a maximal ideal $M$, and then construct another proper ideal $I$ of $R$ so that no proper ideals of $R$ contained $I$ except for $I$ itself? I got stuck at this point for almost an hour, and couldn’t write down something more for that question.

I finally read another half of the question, and understood the statement of the whole question. It would be impossible for me to do the question without knowing the proper ideal that consisted merely of elements of R which don’t have a multiplicative inverse.

After I had proved the statement in that question, I was still not sure whether such a ring exists. I couldn’t give an example of a local ring. This evening, I found it in Wikipedia, and realised that even though I had already convinced myself that a finite field $F$ of prime characteristic $p$ has an algebraic closure $\overline{F} = \bigcup\limits_{i = 1}^\infty \F_{p^i}$, I still didn’t know many properties of the structure of fields.1 The simplest example of a local ring is a field, which has only the trivial ideal $\{0\}$ as its proper ideal.