Rings
Definition
- x + (y + z) = (x + y) + z,
x + y = y + x,
-x + x = 0, 0 + x = x,
x(yz) = (xy)z,
x(y + z) = xy + xz, (x+y)z = xz + yz.
Examples
- Any ring with identity.
- The even integers, under integer addition and multiplication.
- Strictly upper triangular matrices under matrix addition and multiplication
Structure
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Representation
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Decision problems
- Identity problem: Solvable (AC-complete set exists)
- Word problem:Unsolvable
Spectra and growth
- Finite spectrum:
- Free spectrum:
- Growth series:
History/Importance
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References
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Subsystems
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A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu