{
  "id": "1202.0371",
  "version": "v1",
  "published": "2012-02-02T06:22:01.000Z",
  "updated": "2012-02-02T06:22:01.000Z",
  "title": "On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics",
  "authors": [
    "Mahmood Behboodi",
    "Ali Moradzadeh-Dehkordi"
  ],
  "comment": "9 Pages",
  "categories": [
    "math.AC",
    "math.RA"
  ],
  "abstract": "In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \\cal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\\cal{M}}=\\bigoplus_{\\lambda\\in \\Lambda}Rw_{\\lambda}$ and $R/{\\rm Ann}(w_{\\lambda})$ is a principal ideal ring for each $\\lambda \\in \\Lambda$;(3) Every prime ideal of $R$ is a direct sum of at most $|\\Lambda|$ cyclic $R$-modules; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \\cal{M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\\cal{M}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-02T06:22:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13C05",
      "13E05",
      "13F10",
      "13E10",
      "13H99"
    ],
    "keywords": [
      "direct sum",
      "prime ideal",
      "principal ideal",
      "finite direct product",
      "noetherian local"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.0371B"
    }
  }
}