$R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$

Question

Let $R$ be a ring and let $I_1,\dots,I_n,J\subseteq R$ be ideals such that $J\subseteq I_1\cup\dots\cup I_n$. Suppose that there exists an infinite field $K$ and a homomorphism $\phi:K\to R$. Prove that $\exists k$ s.t. $J\subseteq I_k$.

Attempt: (Edited)

If $J\subseteq I_1$ then we're done. Otherwise, let $x\in I_1$ and let $y\in J\setminus I_1$. $\forall r\in K, x+ry\notin I_1$ (because otherwise we'll get $y\in I_1$).

$K$ is infinite so we have infinitely many elements of the form $x+ry\in J$. So there's some $k$ for which $I_k$ contains infintely many elements of the former form.

If $n=2$ so $x+y\notin I_1$ implies $$x+y\in I_2\Rightarrow x\in I_1\Rightarrow I_1\subset I_2\Rightarrow J\subseteq I_2$$

Suppose the theorem holds for some $n$. $x+y\notin I_1\Rightarrow \exists k,I_k\ni x+ry$ for infintely many elements as above.

Answer 1

Here's a solution, it has nothing to do with commutative algebra, this is purely linear algebra. Let's proceed by induction. The case $n=1$, is obvious. I've treated the case $n=2$, in the comment above.

Assume that we have proved the result for $1,..., N-1$, and take $E$ a $k$-vector space, $W_1,...,W_N$ sub vector spaces of $E$ and $V$ a sub vector space contained in $W_1\cup...\cup W_N$.

If $V\subset W_1$ we're done, and if $V\subset W_2\cup...\cup W_N$ we're done too by the induction hypothesis.

So we may assume that we can find $x\in V$ s.t $x\in W_1$, $x\notin W_2\cup...\cup W_N$ and $y\in W_2\cup...\cup W_N$ and $y\notin W_1$.

As $k$ is infinite, the elements $x+ty$ are all in $V$ and there a infintely many of them, so 2 of them must lie in a $W_k$.

Wa have 2 case, if $x+ty$ and $x+sy$ both lie in $W_1$, with $s\neq t$, then $y$ lies in $W_1$ which we excluded. If $x+ty$ and $x+sy$ both lie in $W_k$ for some $k>1$ and $s\neq t$, then certainly $s\neq 0$ and $t\neq 0$ otherwise $x$ would lie in $W_k$, thus $x/s+y$ and $x/t+y$ lie in $W_k$ thus $x.(1/s-1/t)$ lies in $W_k$, which again is excluded.

As ideals are in particular $k$-sub vector spaces of the $k$-algebra $R$, this implies your statement.