Simple Bimodules Over Division Rings: Conditions?

Hey guys! Today, we're diving deep into the fascinating world of noncommutative algebra, specifically focusing on bimodules over division rings. This is a bit of a niche topic, but trust me, it's super interesting, especially if you're into abstract algebra. We'll be tackling a question about the simplicity of bimodules when viewed from both sides, which can seem a little daunting at first, but we'll break it down together.

What are Division Rings and Bimodules?

Before we jump into the heart of the problem, let's make sure we're all on the same page with some key definitions. First off, by a ring, we mean an associative ring with multiplicative unity – that's just fancy math talk for a set with addition and multiplication operations that behave nicely, with a '1' element that doesn't change anything when you multiply by it. Now, a division ring, also known as a skew field, is a special kind of ring where every nonzero element has a multiplicative inverse. Think of the real numbers or complex numbers – every nonzero number has a reciprocal. But division rings can be more exotic; they don't necessarily require multiplication to be commutative (meaning the order matters, so a * b might not equal b * a). This non-commutativity is what makes them so interesting and leads to some surprising results.

Now, what about bimodules? A bimodule is like a vector space, but instead of scalars coming from a field, they come from two different rings. So, if we have two rings, D and S, a (D, S)-bimodule $_DM_S$ is an abelian group M (a set with an addition operation) where we can multiply elements of M on the left by elements of D and on the right by elements of S, and these multiplications play well with each other. More formally, we have left D-module and right S-module structures on M that are compatible, meaning that for any d in D, m in M, and s in S, we have (dm)s = d(ms). This compatibility condition is crucial; it's what makes it a bimodule rather than just two separate module structures.

To truly grasp the essence of bimodules, it's helpful to consider some concrete examples. One classic example is the ring of n x n matrices over a division ring D. This ring naturally has a (D, D)-bimodule structure, where the left and right multiplications are just the usual matrix multiplications. Another example arises from ring homomorphisms. If we have a ring homomorphism f: R -> S, then S can be viewed as an (S, R)-bimodule, where the left multiplication is the usual ring multiplication in S, and the right multiplication is given by s * r = s * f(r)* for s in S and r in R. These examples highlight the versatility of bimodules and their connections to other algebraic structures.

Understanding these foundational concepts is paramount before we delve deeper into the question of simplicity. Without a solid grasp of division rings and bimodules, the nuances of simplicity and its implications can be easily missed. So, take your time, revisit these definitions if needed, and make sure you're comfortable with the basic building blocks. With this groundwork in place, we're well-equipped to tackle the central question.

The Simplicity Question: $_DM_S$ Over Division Rings

Okay, with the basics covered, let's get to the core question. Imagine we have two division rings, D and S, and a (D, S)-bimodule $_DM_S$. The burning question is: what conditions make this bimodule simple when viewed from both sides? What does that even mean, you might ask? Well, simplicity in this context means that the bimodule has no non-trivial submodules. But wait, we need to be specific about what kind of submodules we're talking about.

Since $_DM_S$ is both a left D-module and a right S-module, we can consider two types of submodules: left D-submodules and right S-submodules. A left D-submodule is a subset of M that's closed under addition and left multiplication by elements of D. Similarly, a right S-submodule is closed under addition and right multiplication by elements of S. Now, here's the kicker: for $_DM_S$ to be simple on both sides, it needs to have no non-trivial submodules of either kind. This is a much stronger condition than just being simple as a left module or as a right module individually.

Think of it this way: if $_DM_S$ is simple as a left D-module, it means you can't find a subset that's closed under left multiplication by D (except for the trivial ones – the zero submodule and the whole module itself). But there might still be a right S-submodule lurking in there. To be simple on both sides, we need to ensure that no such submodules exist, neither left nor right. This dual simplicity is a powerful constraint that significantly restricts the structure of the bimodule.

To illustrate this concept, consider a simple example. Let D and S be the same field, say the field of real numbers R. Then, R itself can be considered as an (R, R)-bimodule, where the bimodule structure is just the usual multiplication in R. This bimodule is simple on both sides because any submodule (either left or right) would have to be an ideal of R, and R is a field, which means it has no non-trivial ideals. This example, while simple, gives us a tangible sense of what it means for a bimodule to be simple on both sides.

Now, let's ramp up the complexity a bit. What if D and S are different division rings? Or what if they're the same, but noncommutative? The situation becomes much more intricate. The interplay between the left and right module structures introduces a level of sophistication that requires careful analysis. The existence of non-trivial submodules, or lack thereof, hinges on the relationship between D, S, and the bimodule structure itself.

Exploring this question further, we might ask: are there any general criteria for determining when a bimodule over division rings is simple on both sides? Are there specific classes of bimodules that are guaranteed to have this property? These are the kinds of questions that drive research in this area. Understanding the simplicity of bimodules is crucial for understanding the structure of rings and modules more broadly. It's a fundamental concept that pops up in various contexts, from representation theory to the study of noncommutative rings.

Exploring Necessary and Sufficient Conditions

So, what are some ways we can figure out when a bimodule $_DM_S$ over division rings D and S is simple on both sides? This is where things get really interesting! We need to delve into the properties of D, S, and the bimodule structure itself to uncover the necessary and sufficient conditions for this dual simplicity. One avenue to explore is the concept of balanced bimodules. A bimodule $_DM_S$ is called balanced if the canonical map from $D o End(M_S)$ and $S^{op} o End(_DM)$ are surjective. Here, $End(M_S)$ denotes the ring of S-endomorphisms of M, and $End(_DM)$ denotes the ring of D-endomorphisms of M. This balanced condition turns out to be deeply connected to the simplicity of the bimodule.

Intuitively, the balanced condition ensures that the actions of D and S on M are