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Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ideals are ideals in the polynomial ring generated by monomials. These ideals have several nice properties (e.g. modular law holds (intersection distributes over addition), radicals are generated by squarefree parts of each monomial, integral closure is described by the Newton polyhedron, a Stanley-Reisner ring is a quotient by a monomial ideal, etc to name a few).

In light of this, let us define a pseudo-monomial to be an element in $R$ which is a product of linear homogeneous polynomials in $R$ (with repetition allowed) and a pseudo-monomial ideal to be an ideal in $R$ generated by such elements. I was wondering if these ideals have been studied at all - I would love to see some references.

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The short answer is yes. Thinking geometrically, this is the theory of subspace arrangements. All kinds of things break down, but there are various results in specific cases. See e.g. the unique Bjorner-Peeva-Sidman paper. –  Alexander Woo Nov 18 '10 at 16:25
    
@Alexander: Thanks, this is a great reference. –  Timothy Wagner Nov 19 '10 at 23:34

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