The rings with identity whose additive subgroups are one-sided ideals

ABSTRACT

Let R be a ring with identity. Then {0} and R are the only additive subgroups of R if and only if R is isomorphic (as a ring with identity) to (exactly) one of {0}, $\mathbb{Z}/p\mathbb{Z}$ for a prime number p. Also, each additive subgroup of R is a one-sided ideal of R if and only if R is isomorphic to (exactly) one of {0}, $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$ for an integer n ≥ 2. This note could find classroom use in a first course on abstract algebra as enrichment material for the unit on ring theory.

KEYWORDS:

• Ring with identity
• additive subgroup
• ideal
• prime subring
• Lagrange's theorem
• cyclic group

Disclosure statement

No potential conflict of interest was reported by the author.