Rings with identity / Rings with 1 / Unital rings

Definition

• x + (y + z) = (x + y) + z, x + y = y + x, -x + x = 0, 0 + x = x, x(yz) = (xy)z, x1 = x, 1x = x, x(y + z) = xy + xz, (x+y)z = xz + yz.

Examples

• The integers, Z, under ordinary addition and multiplication.
• The integers modulo n, Z_n under addition and multiplication modulo n.
• n x n matrices over a ring, with matrix addition and multiplication.

Structure

Representation

Decision problems

Identity problem: Solvable (AC-complete set exists)

Word problem:Unsolvable

Spectra and growth

Finite spectrum:

Free spectrum:

Growth series:

History/Importance

References

Subsystems