The Bordalo order on a commutative ring
Melvin Henriksen; Frank A. Smith
Commentationes Mathematicae Universitatis Carolinae (1999)
- Volume: 40, Issue: 3, page 429-440
- ISSN: 0010-2628
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topHenriksen, Melvin, and Smith, Frank A.. "The Bordalo order on a commutative ring." Commentationes Mathematicae Universitatis Carolinae 40.3 (1999): 429-440. <http://eudml.org/doc/248379>.
@article{Henriksen1999,
abstract = {If $R$ is a commutative ring with identity and $\le $ is defined by letting $a\le b$ mean $ab=a$ or $a=b$, then $(R,\le )$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\le )$ to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings $Z_\{n\}$ of integers mod $n$ for $n\ge 2$. In particular, if $R$ is reduced, then $(R,\le )$ is a lattice iff $R$ is a weak Baer ring, and $(R,\le )$ is a distributive lattice iff $R$ is a Boolean ring, $Z_\{3\},Z_\{4\}$, $Z_\{2\}[x]/x^\{2\}Z_\{2\}[x]$, or a four element field.},
author = {Henriksen, Melvin, Smith, Frank A.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {commutative ring; reduced ring; integral domain; field; connected ring; Boolean ring; weak Baer Ring; regular element; annihilator; nilpotents; idempotents; cover; partial order; incomparable elements; lattice; modular lattice; distributive lattice; commutative ring; connected ring; annihilator; nilpotent; residues modulo ; partial order; lattice; modular lattice; distributive lattice},
language = {eng},
number = {3},
pages = {429-440},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Bordalo order on a commutative ring},
url = {http://eudml.org/doc/248379},
volume = {40},
year = {1999},
}
TY - JOUR
AU - Henriksen, Melvin
AU - Smith, Frank A.
TI - The Bordalo order on a commutative ring
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 3
SP - 429
EP - 440
AB - If $R$ is a commutative ring with identity and $\le $ is defined by letting $a\le b$ mean $ab=a$ or $a=b$, then $(R,\le )$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\le )$ to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings $Z_{n}$ of integers mod $n$ for $n\ge 2$. In particular, if $R$ is reduced, then $(R,\le )$ is a lattice iff $R$ is a weak Baer ring, and $(R,\le )$ is a distributive lattice iff $R$ is a Boolean ring, $Z_{3},Z_{4}$, $Z_{2}[x]/x^{2}Z_{2}[x]$, or a four element field.
LA - eng
KW - commutative ring; reduced ring; integral domain; field; connected ring; Boolean ring; weak Baer Ring; regular element; annihilator; nilpotents; idempotents; cover; partial order; incomparable elements; lattice; modular lattice; distributive lattice; commutative ring; connected ring; annihilator; nilpotent; residues modulo ; partial order; lattice; modular lattice; distributive lattice
UR - http://eudml.org/doc/248379
ER -
References
top- Berberian S., Baer-rings, Springer-Verlag, New York, 1972. Zbl0679.16011MR0429975
- Bordalo G., Naturally ordered commutative rings, preprint.
- Speed T., Evans M., A note on commutative Baer rings, J. Austral. Math. Soc. 13 (1971), 1-6. (1971) MR0294318
- Jacobson N., Basic Algebra I, W.H. Freeman and Co., San Francisco, 1974. Zbl0557.16001MR0356989
- Kist J., Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. 13 (1963), 31-50. (1963) Zbl0108.04004MR0143837
- Speed T., A note on commutative Baer rings I, J. Austral. Math. Soc. 14 (1972), 257-263. (1972) MR0318120
- Speed T., A note on commutative Baer rings II, ibid. 15 (1973), 15-21. (1973) MR0330140
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