You now need to expand a knowledge point into a comprehensive professional knowledge structure.\n\nFor example:\n\nKnowledge point: Graphene\nField: Materials Science\nProfessional knowledge structure:\n- Basic properties: Single-layer graphene structure, mechanical properties, electronic properties, thermal properties\n- Preparation methods: Chemical vapor deposition, chemical reduction, mechanical exfoliation, liquid exfoliation\n- Experimental methods: Including material preparation, testing, and characterization techniques\n- Material structure: Describing crystal structure, surface structure, etc.\n- Material performance: Including mechanical, electrical, and thermal performance indicators\n- Material applications: Describing applications and prospects in various fields\n- Material design: Design methods and strategies for new materials, including computational simulations and experimental combinations\n👍\nNow, start outputting the comprehensive professional knowledge structure. Do not explain or annotate. End the output with a 👍 to indicate completion.\n\nKnowledge point: Contents. Introduction. 1. The main theorem: Bases for associative k-algebras. 2. Application: A problem on idempotents. 3. Application: The Poincare-Birkhoff-Witt theorem. 4. Applications: How to take advantage of identities. 5. Observations on some practical questions. 6. Generalization: Bimodule-structures of k-rings. 7. Generalization: Right bases for k-rings. 8. Applications: Coproducts of k-rings. 9. Analogs: Semigroups\nProfession First repeat the request without change, then give your answer.
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