Definition:Ideal of Ring/Left Ideal

This page is about left ideal of ring in the context of ring theory. For other uses, see ideal.

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$.

$J$ is a left ideal of $R$ if and only if:

$\forall j \in J: \forall r \in R: r \circ j \in J$

that is, if and only if:

$\forall r \in R: r \circ J \subseteq J$

Also see

• Definition:Right Ideal of Ring

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Exercise $22.22$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): ideal
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ideal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ideal
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): ideal