<div dir="ltr"><div>Dear list members,</div><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><div class="gmail_quote"><div dir="ltr"><br>You are most cordially invited to the <span><span><span><span><span>Yeditepe Mathematics Department 25th Year Seminar</span></span></span></span></span>s organized by the Department of Mathematics. The details of this week's talk are as follows (please note the unusual time):</div><div dir="ltr"><br></div><div dir="ltr">
Speaker: Engin Büyükaşık (İzmir Institute of Technology)<br></div><div dir="ltr"><br></div><div>Title: Dual Baer Criterion and R-projectivity of injective modules</div><div><br></div><div>Abstract:
Let $R$ be a ring with unity and Mod-$R$ be the category of right $R$-modules. The Baer's Criterion for injectivity states that a right module $M$ is injective iff it is $R$-injective, that is for each right ideal $I$ of $R$, any homomorphism from $I$ into $M$ extends to $R$. Dually, a right module $P$ is $R$-projective if for each right ideal $I$ of $R$ any homomorphism from $M$ into $R/I$ lifts to $R$. Unlike the case for injectivity, $R$-projective modules need not be projective. That is, the Dual Baer Criterion (DBC, for short) does not hold over every ring. The rings $R$ for which the DBC holds in Mod-$R$ are called right testing. From [4], it is known that right perfect rings are right testing. In [3], Faith stated the characterization of all right testing rings as an open problem. Recently in [6], Trlifaj proved that the problem of characterizing right testing rings is undecidable in ZFC. <br> <br> In this talk, after summarizing the aforementioned results, I will mention an extend of the notion of $R$-projectivity, and discuss some problems related to the rings whose injective right modules are $R$-projective which are partially solved in [1].<span style="font-family:CMR12;font-size:12pt;font-style:normal;font-weight:normal;color:rgb(0,0,0)"></span></div><div><span style="font-family:CMR12;font-size:12pt;font-style:normal;font-weight:normal;color:rgb(0,0,0)"><br></span></div><div><span style="font-family:CMR12;font-size:12pt;font-style:normal;font-weight:normal;color:rgb(0,0,0)"></span><font size="2"><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)"><font size="4">References</font><br></span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">[1] Y. Alagöz and E. Büyükaşık, Max-projective modules, </span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">J. Algebra Appl. </span><span style="font-family:CMBX12;font-style:normal;font-weight:bold;color:rgb(0,0,0)">20 </span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">(2021), no. 6. 2150095.<br>[2] H. Alhilali, Y. Ibrahim, G. Puninski, and M. Yousif, When R is a testing module for projectivity?<br></span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">J. Algebra </span><span style="font-family:CMBX12;font-style:normal;font-weight:bold;color:rgb(0,0,0)">484 </span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">(2017), 198-206.<br>[3] C. Faith, </span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">Algebra. II</span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">, Springer-Verlag, Berlin-New York, 1976. Ring theory, Grundlehren der Mathematischen Wissenschaften, No. 191.<br>[4] F .L. Sandomierski, </span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">Relative injectivity and projectivity</span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">, 1964. Thesis (Ph.D.) The Pennsylvania<br>State University.<br>[5] J. Trlifaj, Whitehead test modules, </span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">Trans. Amer. Math. Soc. </span><span style="font-family:CMBX12;font-style:normal;font-weight:bold;color:rgb(0,0,0)">348 </span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">(1996), no. 4, 1521-1554.<br>[6] J. Trlifaj, Faith’s problem on R-projectivity is undecidable, </span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">Proc. Amer. Math. Soc. </span><span style="font-family:CMBX12;font-style:normal;font-weight:bold;color:rgb(0,0,0)">147 </span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">(2019),<br>no. 2, 497-504.<br>[7] J. Trlifaj, The dual Baer Criterion for non-perfect rings, </span><span style="font-family:CMTI12;font-style:italic;font-weight:normal;color:rgb(0,0,0)">Forum Math. </span><span style="font-family:CMBX12;font-style:normal;font-weight:bold;color:rgb(0,0,0)">32 </span><span style="font-family:CMR12;font-style:normal;font-weight:normal;color:rgb(0,0,0)">(2020), no. 3, 663-672.</span></font>
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<div>Date: Friday, May 28, 2021</div>
<div>Time: 16:00</div>
<div>Zoom: Meeting ID: 889 3945 0567<br>Passcode: 7tpSeminar<br><br><br><span></span></div><div><span>--</span></div><div><span><a href="https://researchseminars.org/seminar/7tepemathseminars" target="_blank">https://researchseminars.org/seminar/7tepemathseminars</a><br></span></div><div><br><span></span></div></div></div>
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