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We know that for a finite field $F$, every function on $F$ can be given by apolynomial with coefficients in $F$. What about the converse? i.e. if $R$ is aring (not necessarily commutative or with unity) such that every function on$R$ can be given by a polynomial with coefficients in $R$, can we say $R$ is afinite field ? We show that the answer is yes, and that in fact it is enough toonly require that all bijections be given by polynomials. If we allow our ringsto have unity, we show that the property that all characteristic functions canbe given by polynomials actually characterizes finite fields and if we moreoverallow our rings to be commutative, then to characterize finite fields, it isenough that some special characteristic function be given by a polynomial (withcoefficients even in some extension ring). Motivated by this, we determine allcommutative rings with unity which admits a characteristic function which canbe given by some polynomial with coefficients in the ring.

Classification of Rings which admits some special type of Polynomial functions