rings

# Rings ## When to Use Use this skill when working on rings problems in abstract algebra. ## Decision Tree 1. **Is R a ring?** - (R, +) is an abelian group - Multiplication is associative - Distributive laws: a(b+c) = ab + ac and (a+b)c = ac + bc - `z3_solve.py prove "ring_axioms"` 2. **Ring Properties** - Commutative ring: ab = ba for all a, b? - Ring with unity: exists 1 such that 1*a = a*1 = a? - Integral domain: ab = 0 implies a = 0 or b = 0? - `z3_solve.py prove "integral_domain"` 3. **Ideals** - I is ideal if: I is additive subgroup AND for all r in R, a in I: ra in I, ar in I - Principal ideal: (a) = {ra : r in R} - `sympy_compute.py simplify "r*a"` for ideal multiplication 4. **Ring Homomorphisms** - phi(a + b) = phi(a) + phi(b) - phi(ab) = phi(a)phi(b) - phi(1) = 1 (for rings with unity) ## Tool Commands ### Z3_Ring_Axioms ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll([a,b,c], a*(b+c) == a*b + a*c)" ``` ### Z3_Integral_Domain ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "a*b == 0 implies a == 0 or b == 0" ``` ### Sympy_Ideal ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "r*a" ``` ## Key Techniques *From indexed textbooks:* - [Abstract Algebra] Reading the above equation mod4\(that is, considering this equation in the quotient ring Z/4Z), we must have {2} =2[9}=[9} ons ( io ‘| where the | he? Checking the few saad shows that we must...