The maximal regular ideal of some commutative rings

Osba, Emad Abu, et al. "The maximal regular ideal of some commutative rings." Commentationes Mathematicae Universitatis Carolinae 47.1 (2006): 1-10. <http://eudml.org/doc/249860>.

@article{Osba2006,
abstract = {In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $\mathfrak \{M\} (R)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $\mathfrak \{M\} (R)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse, when $R$ is a product of local rings (e.g., when $R$ is $\mathbb \{Z\}_\{n\}$ or $\mathbb \{Z\}_\{n\}[i]$), when $R$ is a polynomial or a power series ring, and when $R$ is the ring of all real-valued continuous functions on a topological space.},
author = {Osba, Emad Abu, Henriksen, Melvin, Alkam, Osama, Smith, Frank A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {commutative rings; von Neumann regular rings; von Neumann local rings; Gelfand rings; polynomial rings; power series rings; rings of Gaussian integers (mod $n$); prime and maximal ideals; maximal regular ideals; pure ideals; quadratic residues; Stone-Čech compactification; $C(X)$; zerosets; cozerosets; $P$-spaces; von Neumann regular rings; von Neumann local rings; Gelfand rings; polynomial rings; power series rings},
language = {eng},
number = {1},
pages = {1-10},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The maximal regular ideal of some commutative rings},
url = {http://eudml.org/doc/249860},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Osba, Emad Abu
AU - Henriksen, Melvin
AU - Alkam, Osama
AU - Smith, Frank A.
TI - The maximal regular ideal of some commutative rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 1
SP - 1
EP - 10
AB - In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $\mathfrak {M} (R)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $\mathfrak {M} (R)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse, when $R$ is a product of local rings (e.g., when $R$ is $\mathbb {Z}_{n}$ or $\mathbb {Z}_{n}[i]$), when $R$ is a polynomial or a power series ring, and when $R$ is the ring of all real-valued continuous functions on a topological space.
LA - eng
KW - commutative rings; von Neumann regular rings; von Neumann local rings; Gelfand rings; polynomial rings; power series rings; rings of Gaussian integers (mod $n$); prime and maximal ideals; maximal regular ideals; pure ideals; quadratic residues; Stone-Čech compactification; $C(X)$; zerosets; cozerosets; $P$-spaces; von Neumann regular rings; von Neumann local rings; Gelfand rings; polynomial rings; power series rings
UR - http://eudml.org/doc/249860
ER -