I am currently trying to show the equivalence of the following two properties in a unital, countable dimensional algebra $A$ over a field $K$ of characteristic zero.
- The two-sided ideal $I$ is essential in $A$, in that it has non-trivial intersection with any other ideal.
- For every $a \in A$, if $aI = 0$ then $a = 0$.
There is a similar result Murphy's book $C^*$-Algebras and Operator Theory buried in the discussion before Theorem 3.1.8, but the proof depends on the existence of a positive square root; since I am working with algebras in general, I cannot assume that.
My question is, does this equivalence still work in general algebras. If not, what conditions are needed for equivalence?
This is false in general. For instance, let $A=K[t]/(t^2)$. Then the ideal $I=(t)$ is essential (it is the only nonzero proper ideal), but $tI=0$.
However, the implication $(2\Rightarrow 1)$ is true in any ring and the implication $(1\Rightarrow 2)$ is true in any ring with no nonzero nilpotent elements. To prove $(2\Rightarrow 1)$, suppose $I$ satisfies (2) and let $J$ be any nonzero ideal. Then there is some nonzero $a\in J$, and $aI\neq 0$. Since $aI\subseteq I\cap J$, this means $I\cap J\neq 0$. Since $J$ was arbitrary, $I$ is essential.
Now suppose $A$ has no nonzero nilpotent elements. Let $I\subseteq A$ be an essential ideal and let $J=\{a\in A:aI=0\}$. Then $J$ is an ideal, so if $J$ is nontrivial, $I\cap J$ is nontrivial. But if $a\in I\cap J$, then $a^2\in aI=0$. Since $A$ has no nonzero nilpotent elements, this means $I\cap J=0$, and so $J=0$.