The ring of formal power series is a set of sequences of coefficients indexed by non-negative integers, equipped with operations of addition and multiplication defined in a manner similar to polynomials. This ring allows for manipulation of infinite series as if they were polynomials, providing a powerful tool in various areas of mathematics, including localization at prime ideals and local rings. The elements of this ring can be thought of as formal sums that can converge in certain contexts, making them useful for studying local properties of algebraic structures.
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