Specifically, i have an interest in studying idempotent semiring with respect to linearly dependent and linearly independent set of vectors.
This is an important question.
An RG project directly related to this question is
https://www.researchgate.net/project/An-investigation-of-the-asymptotic-behaviour-of-stochastic-generalized-Bellman-equation-solutions-and-problems-of-queueing-system-analysis
See the full project description.
A research area directly related to vector spaces over semirings is the study of vector spaces over supertropical semirings. See, e.g.:
SUPERTROPICAL LINEAR ALGEBRA
https://arxiv.org/pdf/1008.0025.pdf
See Section 2.3, starting on page 5 for the details.
More to the point in the construction of vector spaces over semirings, see
Which semifields are exact?
https://arxiv.org/pdf/1609.09149.pdf
See Section 2, starting on page 3, for a thorough introduction.
See https://en.wikipedia.org/wiki/Semimodule
This is the natural generalization of a vector space to an algebraic structure over a semi-ring. If you check google Scholar for semimodule, you can find literature on the topic.
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