I didn't understand the difference between the two definitions:
I thought the definition of $B[a_1,\ldots,a_n]$ is exactly the one in the item (b), i.e., $B[a_1,\ldots,a_n]=Ba_1+\ldots+Ba_n$.
I need help
Thanks
[Math] What’s the difference between finite and finitely generated algebras
abstract-algebradefinition
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For the purposes of this answer two nonzero vectors ${\bf x}$, ${\bf y}\in{\mathbb R}^d$ are considered as equivalent if there is a $\lambda>0$ such that ${\bf y}=\lambda\,{\bf x}$. An equivalence class is called a direction, and two vectors belonging to the same equivalence class are said to point into the same direction. The unit sphere $S^{d-1}\subset{\mathbb R}^d$ is a set of representatives for this equivalence relation.
In a one-dimensional setting one has just two directions which then are called senses. They are represented by the two points $1$ and $-1$ making up $S^0\subset{\mathbb R}^1$.
The notion of orientation refers to bases of $d$-dimensional real vector spaces $V$. Two bases $(a_i)_{1\leq i\leq d}$ and $(b_i)_{1\leq i\leq d}$ of $V$ are equally oriented when the matrix $T$ relating them has positive determinant. There are exactly two equivalence classes. When there is a distinguished basis of $V$ (e.g. the standard basis $(e_i)_{1\leq i\leq d}$ of ${\mathbb R}^d$) its orientation is usually considered the positive orientation.
An example: When a hyperplane $H\subset V$, $\>0\in H$, is given then a chosen positive orientation in $V$ induces an orientation of $H$ only after a positive normal vector ${\bf n}\perp H$ has been selected. A basis $(a_i)_{1\leq i\leq d-1}$ of $H$ is then positively oriented if $({\bf a}_1,\ldots, {\bf a}_{d-1},{\bf n})$ is a positively oriented basis of $V$.
Generally, I wouldn't say there's a difference, other than one term being favored over the other, in certain fields.
Mathematically, a sequence/list of elements of a set $A$ is just a function from some subset of the natural numbers, to the set $A$. Symbolically, a sequence/list is just a function $f: D \to A$, where $D \subseteq \mathbb{N}$.
Generally sequences are infinitely long; they map all of the natural numbers to elements of $A$. Most lists I encounter are finite, but there's no reason they need to be. The term sequence is almost always referring to the kind of sequence that analysts use (where the items in our sequence are points/subsets of some topological space), while everyone else has to refer to an ordered collection as a list, so people don't think we're talking about analysis.
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An example: the polynomial $k[X]$ is a finitely generated $k$-algebra bit not a finite $k$-algebra.
An alternative, and possibly more illuminating, form of those definitions is the following: a ring $B$ contain a subring $A$ is