# Ring LWE distribution definitions

This may be a stupid question but I've been stuck on parsing these definitions for a while.

I am reading the paper "On Ideal Lattices and Learning with Errors Over Rings" by Lyubashevsky, Peikert, and Regev. I am trying to understand the error distributions they are proposing. In section 3, they define a set $$\mathbb T = K_{\mathbb R}/R^V$$ where $$K$$ is any number field and $$K_{\mathbb R}$$ is $$K \otimes_{\mathbb Q} \mathbb R$$, $$R$$ is its ring of integers, and $$R^V$$ is the dual module of $$R$$. I am a little unsure on how we are defining $$\mathbb T$$.

First, can we view $$R^V$$ as a submodule of $$K_{\mathbb R}$$ by looking at its image under inclusion? ie, are we really supposed to be looking at $$K_{\mathbb R}/\iota(R^V)$$ where $$\iota: K \to K \otimes_{\mathbb Q} \mathbb R$$ sending $$x \mapsto x \otimes 1$$?

Second, what kind of structure are we viewing $$\mathbb T$$ as? ie is $$\mathbb T$$ a $$\mathbb Q$$ vector space? I know $$R^V$$ has a natural $$\mathbb Z$$ module structure as it is a fractional ideal, but it seems weird to view $$K_{\mathbb R}$$ as a $$\mathbb Z$$ module.

Finally in definition 3.1 (their definition of the ring-lwe distribution), they state a ring-lwe sample is of the form $$(a,b = as/q + e \pmod {modR^V})$$? I am struggling to see how to view $$b$$ as an element of $$K_{\mathbb R}$$.

I apologize if these are "obvious" questions and I would really appreciate some clarification.

Thanks!

$$R_q=R/qR, R_q^{\lor}=R^{\lor}/qR^{\lor}$$.
$$R^{\lor}$$is R-module, $$R_q^{\lor}$$ is $$R_q$$-module(define as: $$\forall x\in R,y\in R^{\lor}, (x+qR)(y+qR^{\lor})=xy+qR^{\lor}$$)
$$a\in R_q,s\in R_q^{\lor}$$，So $$as\in R_q^{\lor}$$,and $$as/q \in \frac{1}{q}R^{\lor}/R^{\lor}$$.Since $$\frac{1}{q}R^{\lor} \subset K_{\mathbb{R}}$$,we get $$as/q + e\,mod\,R^{\lor}\in K_{\mathbb{R}}/R^{\lor}$$