Skip to main content
MathOverflow

Return to Question

improved formatting and wording
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Let $\{p_i\}_{i\in I}$$\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $<a, b> \subseteq \cup_{i\in I}p_i$$\langle a, b\rangle$ generated by $a$ and $b$ is contained in the union $\bigcup_{i\in I}\mathfrak{p}_i$, can we deduce that $<a, b> \subseteq p_j $$\langle a, b\rangle \subseteq \mathfrak{p}_j $ for some $j\in I$?

Let $\{p_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If $<a, b> \subseteq \cup_{i\in I}p_i$, can we deduce that $<a, b> \subseteq p_j $ for some $j\in I$?

Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$ generated by $a$ and $b$ is contained in the union $\bigcup_{i\in I}\mathfrak{p}_i$, can we deduce that $\langle a, b\rangle \subseteq \mathfrak{p}_j $ for some $j\in I$?

Source Link
e.r
e.r
  • 51
  • 2

A property of minimal prime ideals in commutative reduced ring

Let $\{p_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If $<a, b> \subseteq \cup_{i\in I}p_i$, can we deduce that $<a, b> \subseteq p_j $ for some $j\in I$?