# Questions tagged [dimension-theory-algebra]

For questions about notions of dimension, rank, or length used in abstract algebra (e.g. Krull dimension, homological dimensions, composition length, Goldie dimension). Questions about dimension of vector spaces, and rank of linear transformations are better placed under the [linear-algebra] tag.

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### Computing the height of an ideal...?

I hope I'm not overbearing in this site. Yes, I'm still struggling. If you can, I have a question about primary decomposition that still needs help, you can find it in my page. Now I wanted to find ...
170 views

### Real-valued dimension

Let $\overline{\mathbb{R}}_{\geq 0} = \mathbb{R}_{\geq 0} \cup \{\infty\}$. Given a commutative ring with unity $R$, $R\operatorname{-Mod}$ denotes the category of $R$-modules. Given $R$-modules $A$ ...
• 904
1 vote
41 views

### misunderstanding on real algebraic varieties

Bochnak-Coste-Roy's book "Real Algebraic Geometry" (1998) is probably the main reference on this subject. I am probably misunderstanding something very fundamental as I can apparently find ...
• 393
1 vote
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• 253
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### Classical Krull Dimension of Commutative Rings

I've been looking at the extension of Krull dimension to non-commutative rings as defined, for example, in On the Krull-Dimension of Left Noetherian Left Matlis-Rings [Krause, Mathematische ...
• 1,363
1 vote
184 views

### Dimension of $\mathcal{O}_X(X-\{P\})$.

Let $X$ be a smooth projective and irreducible curve and $P\in X$. I am asked to show that the dimension (as a $k$-vector space where $k$ is a algebraically closed field) of $\mathcal{O}_X(X-P)$ is ...
• 388
120 views

### Is the dimension of this subspace 1?

"Let $V=M_{2\times2}(\mathbb{R})$ denote the vectors space of all $2\times2$ matrices with real number entries. Determine which of the following subsets are subspaces of $V$. If it is a subspace, ...
1 vote
46 views

### Dimension of a Matrix subspace

What is the dimension and the number of basis vectors for a subspace of 3×3 symmetric matrices? Earlier my professor told us that the dimension and the number of basis vectors for a subspace are the ...
22 views

### About the regular representation of weak hopf algebra

In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
• 11
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### Rank and dimension of matrix in a linear map

Let's say I map a $3 \times 1$ vector $\underline v=(x, y, z)$ by multiplying it with a $3 \times 3$ matrix of rank $2$. Would I be correct in thinking that it transforms all points in 3D space into a ...
107 views

### Suppose U is a subspace of V such that V/U is finite dimensional. Can we say that V is finite dimensional?

Suppose U is a subspace of V such that V/U is finite dimensional. V/U is the quotient sapce, namely the set of all affine subsets of V parallel to U. I think we cannot show that V is finite ...
• 23
1 vote
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• 11
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### What is the dimension of $T$?

Let $U$, $V$ and $W$ be finite dimensional linear spaces (with dimensions $l$, $m$ and $n>0$, respectively) over field $\mathbb{F}$, and let $f\in{\rm Hom}$ $(U,V)$, $g\in{\rm Hom}$ $(U,W)$ such ...
• 515
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### The Dimension of the Solution Set of a Set of Non-affine commutative Polynomial Equations

I am reading the webpage to learn how to determine the dimension of the solution set of a set of polynomial equations. For the first case (the commutative polynomials), it is mentioned that if it is ...
• 541
1 vote
256 views

• 2,011
### Dimension of ring by $\dim_k (m^i/m^{i +1})$ for all $i(2≦i＜∞)$ is the same as the embedding dimension?
Let $A$ be a Noetherian local ring. Define $m$ be it's maximal ideal and $A/m$ be residue field. Then we can define embedding dimension of $A$ by $\dim_k(m/m^2)$, here $\dim_k m/m^2$ is dimension of \$...