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Problem 1619

National Olympiad, first round

Algebra Difficulty 6.2 Prove it Romanian Mathematical Olympiad · Romania

We say that a ring $A$ has property (P) if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element.

a) If in $A$ , $1 + 1 \neq 0$ , prove that $A$ has property (P) if and only if $A$ is a field.

b) Give an example of a ring that is not a field, containing at least two elements, and having property (P).

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

a) If $A$ is a field and $x \in A$ , $x \ne 0$ , then $x$ is invertible and $x = x + 0$ ; this representation is unique, since $0$ is the only noninvertible element.

Assume $A$ is not a field. Let $x \in A$ , $x \ne 0$ , be a noninvertible element. Since $1 + x = (1 + x) + 0$ and $-1 + x = (-1 + x) + 0$ , the elements $1 + x$ and $-1 + x$ are noninvertible, otherwise the uniqueness of the representation would imply $x = 0$ .

Since $x = 1 + (-1 + x) = -1 + (1 + x)$ , it follows $1 = -1$ , contradiction.

b) Let $E$ be a nonempty set, and $A = (\mathcal{P}(E), \Delta, \cap)$ be the Boolean ring of the set of parts of $E$ . In this ring, $E$ is the only invertible element.

If $X$ is a nonempty part of $E$ , then the element $E \setminus X$ is noninvertible, and $X = E \Delta (E \setminus X)$ . The representation is unique, since $E$ is the only invertible element.

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