Math 2.0

Associative rings and algebras Definition

Two-sided ideal

Builds on Ideal

Definition

Let RR be a ring. A two-sided ideal of RR is a subset I⊆RI \subseteq R with these properties.

  1. II is a subgroup of the additive group (R,+)(R, +).
  2. For all r∈Rr \in R and a∈Ia \in I, ra∈Ir a \in I and ar∈Ia r \in I.

So a two-sided ideal is a left ideal that is also a right ideal.

Remarks

In a commutative ring, left ideals, right ideals and two-sided ideals are the same. The kernel of every ring homomorphism is a two-sided ideal. The quotient ring R/IR/I exists exactly when II is a two-sided ideal. In the ring of n×nn \times n matrices over a field, the only two-sided ideals are {0}\{0\} and the whole ring.

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  1. Group concept
  2. Ring definition
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  4. Two-sided ideal definition

Where this leads

48 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ideal_(ring_theory), Definitions
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