## Exercises in Basic Ring TheoryEach undergraduate course of algebra begins with basic notions and results concerning groups, rings, modules and linear algebra. That is, it begins with simple notions and simple results. Our intention was to provide a collection of exercises which cover only the easy part of ring theory, what we have named the "Basics of Ring Theory". This seems to be the part each student or beginner in ring theory (or even algebra) should know - but surely trying to solve as many of these exercises as possible independently. As difficult (or impossible) as this may seem, we have made every effort to avoid modules, lattices and field extensions in this collection and to remain in the ring area as much as possible. A brief look at the bibliography obviously shows that we don't claim much originality (one could name this the folklore of ring theory) for the statements of the exercises we have chosen (but this was a difficult task: indeed, the 28 titles contain approximatively 15.000 problems and our collection contains only 346). The real value of our book is the part which contains all the solutions of these exercises. We have tried to draw up these solutions as detailed as possible, so that each beginner can progress without skilled help. The book is divided in two parts each consisting of seventeen chapters, the first part containing the exercises and the second part the solutions. |

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### Contents

Zero Divisors | 15 |

Division Rings | 31 |

Socle and Radical | 45 |

Semisimple Rings | 49 |

Prime Ideals Local Rings | 53 |

Polynomial Rings | 59 |

Rings of Quotients | 63 |

Rings of Continuous Functions | 67 |

Ring Homomorphisms | 107 |

Characteristics | 111 |

Divisibility in Integral Domains | 115 |

Division Rings | 121 |

Automorphims | 127 |

The Tensor Product | 133 |

Artinian and Noetherian Rings | 139 |

Socle and Radical | 145 |

Special Problems | 73 |

SOLUTIONS | 77 |

Fundamentals | 79 |

Ideals | 91 |

Zero Divisors | 101 |

Semisimple Rings | 153 |

Prime Ideals Local Rings | 159 |

Polynomial Rings | 169 |

Special problems | 187 |

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### Common terms and phrases

artinian ring Aſh automorphism Boole ring called canonical Chapter char(K clearly commutative ring consider Conversely deduce defined denote direct summand divides division ring easily checked field finite function holds homomorphism f idempotent element implies inclusion injective integral domain inverse ker(f left artinian left ideal left noetherian left right Let f map f matrices maximal ideal minimal Mn(R modules multiplicative system nilpotent elements noetherian ring non-zero divisors obtain obviously phism polynomial previous exercise prime ideal prime number principal ideal proper ideal prove quotient ring R-module rad(R resp right ideal right noetherian right quasi-regular ring homomorphism ring isomorphism ring of quotients ring with identity semisimple ring Show simple Spec(R subgroup subring subring with identity subset surjective surjective ring homomorphism topology unique verify z-filter zero divisor