Previous |  Up |  Next

# Article

Full entry | PDF   (0.2 MB)
Keywords:
essential ideals; $sd$-ideal; almost locally compact space; nowhere dense; Zariski topology
Summary:
Let $\mathcal{Z(R)}$ be the set of zero divisor elements of a commutative ring $R$ with identity and $\mathcal{M}$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq \mathcal{Z(R)}$ and $I$ is contained in no minimal prime ideal. We denote by $R_{K}(\mathcal{M})$, the set of all $a\in R$ for which $\overline{D(a)}= \overline{\mathcal{M}\setminus V(a)}$ is compact. We show that $R$ has property $(A)$ and $\mathcal{M}$ is compact if and only if $R$ has no $sd$-ideal. It is proved that $R_{K}(\mathcal{M})$ is an essential ideal (resp., $sd$-ideal) if and only if $\mathcal{M}$ is an almost locally compact (resp., $\mathcal{M}$ is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring $R$ need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring $R$ is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of $C(X)$ is equal to the socle of C(X) (i.e., $C_{F}(X)= O^{\beta X\setminus I(X)}$). Finally, we show that a topological space $X$ is pseudo-discrete if and only if $I(X)=X_{L}$ and $C_{K}(X)$ is a pure ideal.
References:
[1] Abu Osba E.A., Al-Ezeh H.: Purity of the ideal of continuous functions with compact support. Math. J. Okayama Univ. 41 (1999), 111–120. MR 1816622 | Zbl 0973.54020
[2] Aliabad A.R., Azarpanah F., Taherifar A.: Relative $z$-ideals in commutative rings. Comm. Algebra 41 (2013), 325–341. DOI 10.1080/00927872.2011.630706 | MR 3010540 | Zbl 1264.13003
[3] Azarpanah F.: Intersection of essential ideals in $C(X)$. Proc. Amer. Math. Soc. 125 (1997), 2149–2154. DOI 10.1090/S0002-9939-97-04086-0 | MR 1422843 | Zbl 0867.54023
[4] Azarpanah F.: Essential ideals in $C(X)$. Period. Math. Hungar. 31 (1995), 105–112. DOI 10.1007/BF01876485 | MR 1609417 | Zbl 0867.54023
[5] Azarpanah F., Taherifar A.: Relative $z$-ideals in $C(X)$. Topology Appl. 156 (2009), 1711–1717. MR 2521707 | Zbl 1167.54005
[6] Dietrich W.: On the ideal structure of $C(X)$. Trans. Amer. Math. Soc. 152 (1970), 61–77; MR 42:850. MR 0265941 | Zbl 0205.42402
[7] Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York-Heidelberg, 1976. MR 0407579 | Zbl 0327.46040
[8] Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115 (1965), 110–130. DOI 10.1090/S0002-9947-1965-0194880-9 | MR 0194880 | Zbl 0147.29105
[9] Henriksen M., Woods R.G.: Cozero complemented spaces; when the space of minimal prime ideals of a $C(X)$ is compact. Topology Appl. 141 (2004), 147–170. DOI 10.1016/j.topol.2003.12.004 | MR 2058685 | Zbl 1067.54015
[10] Huckaba J.A.: Commutative Rings with Zero Divisors. Marcel Dekker Inc., New York, 1988. MR 0938741 | Zbl 0637.13001
[11] Huckaba J.A., Keller J.M.: Annihilation of ideals in commutative rings. Pacific J. Math. 83 (1979), 375–379. DOI 10.2140/pjm.1979.83.375 | MR 0557938 | Zbl 0388.13001
[12] Karamzadeh O.A.S., Rostami M.: On the intrinsic topology and some related ideals of $C(X)$. Proc. Amer. Math. Soc. 93 (1985), no. 1, 179–184. MR 0766552 | Zbl 0524.54013
[13] Levy R.: Almost P-spaces. Canad. J. Math. 2 (1977), 284–288. DOI 10.4153/CJM-1977-030-7 | MR 0464203 | Zbl 0342.54032
[14] McConnel J.C., Robson J.C.: Noncommutative Noetherian Rings. Wiley-Interscience, New York, 1987; MR 89j:16023. MR 0934572
[15] Safaean S., Taherifar A.: $d$-ideals, $fd$-ideals and prime ideals. submitted.
[16] Taherifar A.: Some generalizations and unifications of $C_{K}(X)$, $C_{\psi}(X)$ and $C_{\infty}(X)$. arXiv:$1302.0219$ [math.GN].
[17] Veksler A.I.: $P'$-points, $P'$-sets, $P'$-spaces. A new class of order-continuous measures and functions. Soviet Math. Dokl. 14 (1973), 1445–1450. MR 0341447

Partner of