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## On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite)

#### Somayeh Soltanpour [1]

Let $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). We introduce and study  the concept of colocally socle of $C(X)$ as $C_{\mu}{S_{\lambda}}(X)=\left\{ f\in C(X):|X\backslash {S}^{\lambda}_{f}|<\mu \right\}$, where ${{S}^{\lambda}_{f}}$ is the union of all open subsets $U$ in $X$ such that $|U\backslash Z(f)|<\lambda$. $C_{\mu}{S_{\lambda}}(X)$ is a $z$-ideal of $C(X)$ containing ${{C}_{F}}(X)$. In particular, $C_{{\aleph}_0}{S_{{\aleph}_0}}(X)=CC_F(X)$ and $C_{{\aleph}_1}{S_{{\aleph}_1}}(X)=CS_c(X)$ are investigated. For each of the containments in the chain ${{C}_{F}}(X)\subseteq CC_F(X)\subseteq C_{\mu}{S_{\lambda}}(X)\subseteq C(X)$, we characterize the spaces $X$ for which the containment is actually an equality. We determine the conditions such that $CC_F(X)$ ($CS_c(X)$) is not prime in any subrings of $C(X)$ which contains the idempotents of $C(X)$. The primeness of $CC_F(X)$ in some subrings of $C(X)$ is investigated.
functionally countable space, socle, locally socle, colocally socle, prime ideal
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Primary Language en Mathematics Mathematics Orcid: 0000-0002-1072-9845Author: Somayeh Soltanpour (Primary Author) Publication Date : October 8, 2019
 Bibtex @research article { hujms629905, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, volume = {48}, pages = {1430 - 1436}, doi = {}, title = {On the locally socle of \$C(X)\$ whose local cozeroset is cocountable (cofinite)}, key = {cite}, author = {Soltanpour, Somayeh} } APA Soltanpour, S . (2019). On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite). Hacettepe Journal of Mathematics and Statistics , 48 (5) , 1430-1436 . Retrieved from https://dergipark.org.tr/en/pub/hujms/issue/49321/629905 MLA Soltanpour, S . "On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite)". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1430-1436 Chicago Soltanpour, S . "On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite)". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1430-1436 RIS TY - JOUR T1 - On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite) AU - Somayeh Soltanpour Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1430 EP - 1436 VL - 48 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite) %A Somayeh Soltanpour %T On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite) %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 5 %R %U ISNAD Soltanpour, Somayeh . "On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite)". Hacettepe Journal of Mathematics and Statistics 48 / 5 (October 2019): 1430-1436 . AMA Soltanpour S . On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite). Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1430-1436. Vancouver Soltanpour S . On the locally socle of $C(X)$ whose local cozeroset is cocountable (cofinite). Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1436-1430.