## Linear forms on free modules over certain local ring.(English)Zbl 0810.13006

Linear forms $$\varphi : M \to A$$ on a free module $$M$$ over the algebra $$A = \mathbb{R} [x]/x^ m$$ are studied. In particular, it is proved that $$\text{Ker} \varphi$$ is a hyperplane of $$M$$ if and only if it exists $$y \in M$$ such that $$\varphi (y) \notin \eta A$$, where $$\eta$$ is the element of $$A$$ corresponding to $$x$$.

### MSC:

 13C10 Projective and free modules and ideals in commutative rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 15A63 Quadratic and bilinear forms, inner products

### Keywords:

linear forms on a free module
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### References:

 [1] Atiyah M.F., MacDonald I.G.: Introducion to commutative algebra. (Russian), Mir, Moscow, 1972. [2] McDonald B.R.: Geometric algebra over local rings. Pure and applied mathematics, New York, 1976. · Zbl 0346.20027
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