Can Any Non-Commutative Ring Without Zero Divisors Be Embedded into a Division Ring?
The question of whether any non-commutative ring without zero divisors can be embedded into a division ring is both intriguing and pertinent in algebraic theory. This article delves into the conditions and processes involved, providing a comprehensive overview of the topic, and elaborating on relevant mathematical principles.
Understanding Non-Commutative Rings and Division Rings
Non-Commutative Rings are algebraic structures where the multiplication operation is not commutative, i.e., for some elements (a) and (b), it holds that (ab eq ba). On the other hand, a Division Ring, also known as a skew field, is a ring in which every non-zero element has a multiplicative inverse. A fundamental example of a non-commutative division ring is the Quaternion Algebra, which extends the real numbers to include non-commutative multiplication.
Embedding into a Division Ring
The process of embedding a non-commutative ring without zero divisors into a division ring involves constructing a suitable field of fractions. A non-commutative ring without zero divisors is often referred to as a domain. The construction typically involves forming equivalence classes of fractions, akin to the field of fractions of a commutative domain. This construction leads to the creation of a division ring that contains the original ring as a subring.
Key Conditions and Examples
While not every non-commutative domain can be embedded into a division ring, this embedding is possible under certain conditions. For instance, a division ring can be constructed if the non-commutative domain satisfies the Ore condition. The Ore condition is a set of algebraic criteria that, when satisfied, ensure the embeddability of the domain into a division ring. There are two types of conditions: the Right Ore Condition and the Left Ore Condition.
Mathematical Insights and Limitations
While the field of fractions approach works well for commutative rings, the situation becomes more complex for non-commutative rings. An example due to Malcev (1937) demonstrates a non-commutative ring without zero divisors that cannot be embedded into a division ring. This highlights the structural limitations inherent in non-commutative algebra. However, mathematicians have not given up on finding conditions under which such embeddings are possible.
Further Exploration and Conditions
Research in this area has led to the formulation of the Ore Condition by ?ystein Ore in 1931. This condition provides a framework for determining when a non-commutative ring without zero divisors can be embedded into a division ring. Despite the complexity, theorems and constructions based on the Ore condition have advanced our understanding of these algebraic structures significantly.
Conclusion
In summary, while the embedding of non-commutative rings without zero divisors into division rings is not always possible due to structural limitations, theorems and constructions based on the Ore condition provide a path for such embeddings under specific conditions. This exploration not only deepens our understanding of non-commutative algebra but also highlights the ongoing challenges and advancements in modern algebra.