Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb{F}$-vector space $V$ and $W$ a totally isotropic subspace. Then $\operatorname{dim}W \leq \frac{\operatorname{dim}V}{2}$.
I am trying to generalise a result for vector spaces to the case of modules over rings, and the proof of the former uses the above inequality to bound the dimension of a certain subspace, so I was wondering whether the analogous result exists for bilinear forms over rings (e.g. something bounding the length of a totally isotropic submodule in terms of the length of the ambient module). I don't have much experience with bilinear forms over rings, but the bibliography I found online does not mention a similar result.
It is possible that the notion of totally isotropic is substantially different over a ring due to the existence of zero-divisors, however I would be glad if someone could point out a related result (maybe dependent on some properties of the ring) or explain why such an inequality fails to hold over a ring.
Edit: I am reading this paper and wondering whether some of its results can be generalised to rings. My question refers to Lemma 7 (page 7, towards the end), more specifically the sentence starting with "Applying $(2)$ to $A=J_r$...". He proceeds to use the result I mentioned about vector spaces to obtain an inequality, and i was just wondering if the analogous reasoning can be used in a ring setting.