# Product of a principal proper ideal by itself

Let $P$ be a principal proper ideal in an integral domain.

Is it $P^2 \subset P$ in general? If yes, how to prove it?

For example, if you look at the ideal $(3)=3\mathbb{Z}$ in $\mathbb{Z}$, it is quite simple to show that $3$ is not in $(3)^2$, but how to prove a similar property in a more general context?
Now suppose that we have a Dedekind domain.

If $P$ is a principal prime ideal, does $P^2 \neq P$ (and so $P^2 \subset P$) follow from the fact that there is a sort of unique factorization of proper ideals in terms of prime ideals?

-

4
By $\subset$ you mean $\subsetneq$ (gotten from `\subsetneq`). It is usually a good idea to be emphatic about this in notation. – Mariano Suárez-Alvarez♦ Feb 15 '12 at 6:15
Hint: contains = divides for principal ideals and ideals in Dedekind domains. – Math Gems Feb 15 '12 at 6:27

## 1 Answer

If $P=(x)$, then $P^2=(x^2)$. If $P\subseteq P^2$, there is an $a$ such that $x=ax^2$, so $x(1-ax)=0$. If $x\neq0$, then $1-ax=0$ and $x$ is a unit.

Therefore $P=P^2$ implies either $P=0$ or $P=A$, the whole ring.

-

Nice one. And if we replace "principal" with "prime", does the property still hold? – Oo3 Feb 15 '12 at 18:44
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 23, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8961783051490784, "perplexity_flag": "head"}