{
  "id": "1704.01403",
  "version": "v1",
  "published": "2017-04-05T13:21:51.000Z",
  "updated": "2017-04-05T13:21:51.000Z",
  "title": "Some algebraic observations on the ring of periodic distributions",
  "authors": [
    "Amol Sasane"
  ],
  "comment": "17 pages. arXiv admin note: substantial text overlap with arXiv:1606.04685",
  "categories": [
    "math.FA",
    "math.AC"
  ],
  "abstract": "The set of periodic distributions, with usual addition and convolution, forms a ring, which is isomorphic, via taking a Fourier series expansion, to the ring ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ of sequences of at most polynomial growth with termwise operations. In this article we make the following observations: ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ is not Noetherian; ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ is a B\\'ezout ring; ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ is coherent; ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ is a Hermite ring; ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ is not projective-free; ${\\mathcal{S}}'({\\mathbb{Z}}^d)$ is a pre-B\\'ezout ring; for all $m\\in {\\mathbb{N}}$, $SL_m({\\mathcal{S}}'({\\mathbb{Z}}^d))$ is generated by elementary matrices, that is, $SL_m({\\mathcal{S}}'({\\mathbb{Z}}^d))=E_m({\\mathcal{S}}'({\\mathbb{Z}}^d))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-05T13:21:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46H99",
      "13J99"
    ],
    "keywords": [
      "periodic distributions",
      "algebraic observations",
      "fourier series expansion",
      "polynomial growth",
      "usual addition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}