Let's enlarge the ring R, by adjoining a new element & such that ² = 0. In this way, we'll obtain a new ring R[ɛ]. (a) Explain why every element of R[s] can be written in the form a + be, where a, b E R. (b) Given two elements a + be €R[ɛ] and c + de € R[e], write down a formula for their product: (a + b)(c + de) Make use of the relation 2 = 0 to simplify your result. (c) Show that the element 1+ is a unit in R[ɛ], by finding its multiplicative inverse.

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