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Keywords:
commutative ring; reduced ring; integral domain; field; connected ring; \linebreak Boolean ring; weak Baer Ring; regular element; annihilator; nilpotents; idempotents; cover; partial order; incomparable elements; lattice; modular lattice; distributive lattice
commutative ring; reduced ring; integral domain; field; connected ring; \linebreak Boolean ring; weak Baer Ring; regular element; annihilator; nilpotents; idempotents; cover; partial order; incomparable elements; lattice; modular lattice; distributive lattice
Summary:
If $R$ is a commutative ring with identity and $\leq$ is defined by letting $a\leq b$ mean $ab=a$ or $a=b$, then $(R,\leq)$ is a partially ordered ring. Necessary and sufficient conditions on $R$ are given for $(R,\leq)$ 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 $n\geq2$. In particular, if $R$ is reduced, then $(R,\leq)$ is a lattice iff $R$ is a weak Baer ring, and $(R,\leq)$ 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.
References:
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[Bo] Bordalo G.: Naturally ordered commutative rings. preprint.
[ES] Speed T., Evans M.: A note on commutative Baer rings. J. Austral. Math. Soc. 13 (1971), 1-6. MR 0294318
[J] Jacobson N.: Basic Algebra I. W.H. Freeman and Co., San Francisco, 1974. MR 0356989 | Zbl 0557.16001
[K] Kist J.: Minimal prime ideals in commutative semigroups. Proc. London Math. Soc. 13 (1963), 31-50. MR 0143837 | Zbl 0108.04004
[Sp1] Speed T.: A note on commutative Baer rings I. J. Austral. Math. Soc. 14 (1972), 257-263. MR 0318120
[Sp2] Speed T.: A note on commutative Baer rings II. ibid. 15 (1973), 15-21. MR 0330140
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