# Are integral extensions of a catenary ring still catenary?

A (commutative unitary) Noetherian ring $$R$$ of finite dimension is said to be catenary if for every prime ideal $$\mathfrak{p}$$ of $$R$$ one has $$\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\mathrm{dim}R$$.

Let $$A$$ be a catenary ring and $$B$$ a finite extension of $$A$$ (id est $$A\subseteq B$$ and $$B$$ is a finite $$A$$-module). Is $$B$$ a catenary ring?

• Is your definition of catenary equivalent to the usual one (that requires that for all primes $p \subsetneq q$ all maximal prime chains $p \subsetneq \cdots \subsetneq q$ have the same length)? – tj_ Mar 24 '19 at 10:25
{"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": 1, "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, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9666882753372192, "perplexity": 168.2687634610307}, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107874340.10/warc/CC-MAIN-20201020221156-20201021011156-00473.warc.gz"}