Crosscap two characterization of dot product graphs of commutative rings
Bibliographic record
Abstract
<p>Let <span class="math inline">\(A\)</span> be a commutative ring with nonzero identity and <span class="math inline">\(n\geq 2\)</span> be a positive integer. With the ring <span class="math inline">\(R=A\times\cdots\times A\)</span> (<span class="math inline">\(n\)</span> times), one can associate graphs <span class="math inline">\(TD(R)\)</span> and <span class="math inline">\(ZD(R)\)</span> respectively called the total dot product graph and the zero-divisor dot product graph of <span class="math inline">\(R\)</span>. In this paper, we study some topologicaal properties of these two dot product graphs of <span class="math inline">\(R.\)</span> In particular, it is shown that, the zero-divisor dot product graph <span class="math inline">\(ZD(R)\)</span> is a projective graph if and only if <span class="math inline">\(R\)</span> is isomorphic to <span class="math inline">\(\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}\times\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}.\)</span> Moreover, we prove that no total dot product graph can be projective. With these observations, we classify all commutative rings for which dot product graphs <span class="math inline">\(ZD(R)\)</span> and <span class="math inline">\(TD(R)\)</span> have crosscap two.</p>
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".