- #1
jdinatale
- 155
- 0
Maybe I'm misinterpreting the question, I'm not sure how to prove that n_0 i = 0.
micromass said:I don't get why you multiply both on the left and on the right. I would think that all modules here are left R-modules. So you should always multiply with R on the left. In particular, we have
[tex]A=\{m\in M~\vert~im=0~\text{for all}~i\in R\}[/tex]
and so on.
micromass said:And what are X and Y?
The annihilator in abstract algebra is a concept used to describe the set of elements in a ring that, when multiplied by a specific element, result in the identity element (typically denoted as 1). It is denoted as Ann(a) and can also refer to the set of elements that are mapped to 0 under a given ring homomorphism.
Proving basic facts about the annihilator is important because it helps to understand the properties and behavior of this concept. It also provides a deeper understanding of rings, modules, and other algebraic structures, which have applications in various fields such as physics, computer science, and cryptography.
To prove a basic fact about the annihilator in abstract algebra, one typically uses the definition of the annihilator, along with other properties and theorems in abstract algebra. This may involve using techniques such as proof by contradiction, induction, or direct proof.
Some common basic facts about the annihilator in abstract algebra that are frequently proven include the fact that the annihilator of a ring element is an ideal, that the annihilator of a submodule is a submodule, and that the annihilator of a ring is a maximal ideal.
Yes, there are several practical applications of proving basic facts about the annihilator in abstract algebra. For example, it is used in coding theory to construct error-correcting codes, in cryptography to create secure encryption algorithms, and in signal processing to analyze and process signals.