To address the question of whether the characteristic of a simple ring is either prime or zero, we must first clarify some definitions and properties related to rings and their characteristics. This article will explore the proof of this statement, providing a thorough understanding of simple rings and their characteristics.

Definitions

Simple Ring: A ring (R) is called simple if it has no non-trivial two-sided ideals other than ({0}) and (R) itself. This implies that (R) is non-zero and has no proper ideals.

Characteristic of a Ring: The characteristic of a ring (R), denoted (text{char}R), is the smallest positive integer (n) such that (n cdot 1_R 0), where (1_R) is the multiplicative identity in (R). If no such (n) exists, the characteristic is defined to be zero.

Proving the Statement

To prove that the characteristic of a simple ring is either prime or zero, we can follow these steps:

Step 1: Assume (text{char}R n) is a Positive Integer

Assume that the characteristic of (R) is a positive integer (n). This means (n cdot 1_R 0).

Step 2: Consider the Ideal Generated by (1_R)

Consider the ideal generated by (1_R). Since (R) is a ring with unity, the ideal generated by (1_R) is the entire ring (R). Therefore, any element in (R) can be expressed as (r cdot 1_R) for some (r in R).

Step 3: Analyze the Equation (n cdot 1_R 0)

Since (R) is simple, if (n) is not prime then (n) can be expressed as (ab) for some integers (a, b > 1). This leads to the following analysis:

Consider the Element (a cdot 1_R): Since (a, b > 1), the ideal generated by (a cdot 1_R) is non-trivial and not equal to (R) or ({0}). Similarly, (b cdot 1_R): Similarly, the ideal generated by (b cdot 1_R) generates a non-trivial ideal. Non-Trivial Ideals Contradict the Simplicity of (R): Thus, we have non-trivial ideals (a cdot 1_R) and (b cdot 1_R), contradicting the simplicity of (R).

Conclusion for Positive Characteristic

Therefore, if (text{char}R n) is a positive integer, (n) must be prime.

When (text{char}R 0)

If (R) has characteristic zero, it means that no positive integer (n) satisfies (n cdot 1_R 0). This is also a valid case and does not contradict the simplicity of the ring.

Final Result

From this analysis, we conclude that:

The characteristic of a simple ring is either zero or a prime number.

This result aligns with the structure of simple rings and their ideals, confirming the statement you are interested in proving or disproving.