Solution: Kaplansky’s Commutative Rings 1.1.1

In an effort to keep this blog active, I’ve decided to post some old solutions to various problems I’ve done from books, starting with Kaplansky’s “Commutative Rings”. Hopefully they might be useful to someone learning the subject. I’ve done about half the problems, so it might take a while to post them all, and perhaps posting them will motivate me to do the other half. I won’t necessarily post them all in order, but I will start with the first problem in the book. I encourage readers to try the problem first, and post alternative solutions in the comments! The wording is changed slightly to avoid copyright issues, and I will attempt to provide some background comments to make the post self-contained. All the solutions will be linked to in the Solutions Page for easy reference.

Problem. If every proper ideal in a commutative ring $R$ is prime, show that $R$ is a field.

Solution. Let $x\in R$ be nonzero. Then $(x^2)$ is a prime ideal, so $x\in (x^2)$. Hence, there is a $c\in R$ such that $cx^2 = x$, or equivalently, $x(cx-1) = 0$. However, $(0)$ is also a prime ideal, so that $R$ is an integral domain. Since $x\not=0$, we must have $cx – 1 =0$, so that $x$ is invertible.

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