Wisconsin inmate Nathaniel Lindell seeks damages under 42 U.S.C. § 1983, claiming that his First Amendment rights were violated when his copy of Issue # 45 of the magazine Pagan Revival was seized by prison officials and subsequently “lost.” This case returns to us for a second time. The matter came before us the first time on appeal of the district court’s dismissal of Lindell’s original complaint during the screening phase. See 28 U.S.C. § 1915A(b)(l). We affirmed the district court’s decision in most respects but remanded for consideration of Lindell’s First Amendment freedom-of-speeeh claim, which we found sufficient to satisfy the notice pleading standard. The case now returns as an appeal from the district court’s grant of summary judgment to the defendant. We affirm.
Lindell was at all relevant times an inmate at Waupun Correctional Institution (WCI). WCI is a maximum security prison that houses approximately 1,240 inmates and receives thousands of pieces of mail a day. According to Lindell, Pagan Revival is a “magazine which is Euro-centric, and discusses the struggles, thoughts and goals of the Euro-pagan community.” Lindell claims that Issue # 45 was sent to him at the prison and was received by the security office for screening on July 25, 2000, but that it disappeared shortly thereafter. Warden McCaughtry says that there is no record of the issue having passed through security, and Lindell was never notified that the delivery was refused, as required by prison regulations. See Wis. Admin. Code § DOC 309.05(3). When Lindell contacted the security director to find out what happened to the magazine in October 2000, he was told that the magazine was lost, and that his only recourse was to file a complaint. In response to his complaint, the prison reimbursed Lindell for the missing issue. He was also warned that the publication was “deemed inappropriate” and that he would not be allowed to possess it at WCI. See Wis. Admin. Code § DOC 309.05(2)(b)(l) (banning publications that “[t]each or advocate violence or hatred and present a danger to institutional security and order”). Although security did not have Lindell’s copy, they did have a record of screening and rejecting another copy of *875the same issue requested by another inmate. After exhausting his administrative remedies, Lindell filed his complaint with the district court.
The district court initially denied his request for leave to proceed in forma pauperis and dismissed his complaint under § 1915A(b)(l) because it found that the claims were frivolous. Lindell filed a Rule 59(e) motion and in the alternative requested to submit an amended and supplemented complaint pursuant to Rule 15(a). When the district court denied the Rule 59(e) motion, Lindell appealed. We affirmed in most respects but vacated the dismissal of his freedom-of-speech claim. See Lindell v. Doe, 58 Fed.Appx. 638 (7th Cir.2003).
Lindell then requested leave to file an amended complaint, which the district court denied, but it permitted him to “proceed against [two] Doe defendants” if he could identify and serve them, and, “in order to assist plaintiff in identifying the two Doe defendants,” the court amended the complaint to name McCaughtry as a defendant on its own motion. McCaughtry was to be replaced as defendant when the Does were identified. At the pretrial conference, the district court ordered McCaughtry to help Lindell identify the persons in the mailroom and the security office who received the magazine. McCaughtry stated that the best he could do was to list the persons scheduled to work in the mailroom and the security supervisor’s office on the relevant dates. Lindell moved for contempt and Rule 11 sanctions and requested a court order compelling McCaughtry to identify the persons who handled the magazine.
The district court denied Lindell’s motions and dismissed the unidentified mail-room employees, stating that “plaintiffs obsession with learning precisely who lost or intentionally took his magazine is needlessly delaying the progress of this lawsuit.” The court reasoned that “[b]ecause plaintiff was told that he would not be allowed to receive issue #45 under any circumstances, it is irrelevant who lost or took the publication.” It refused a second request by Lindell to amend his complaint because adding new defendants and claims would unduly delay the proceedings, and because any amendment that challenged the DOC regulation itself — rather than just its application — would be futile. The district court also denied Lindell’s motion to obtain a copy of Issue #45 because allowing a plaintiff to obtain a magazine determined to threaten institutional security through the discovery process “would render the institution’s review system superfluous and would encourage inmates to file lawsuits as a way to circumvent the institution’s security procedures.”
After moving for summary judgment, McCaughtry attempted to file supplemental proposed findings of facts and additional affidavits to strengthen his assertion that the refusal to allow Lindell to possess Issue #45 was reasonably related to a legitimate penological interest. The district court struck most of the submission as untimely, but it did accept three of the defendant’s proposed findings of fact, which it considered responsive to a statement in Lindell’s opposition to defendant’s motion for summary judgment. A motion by Lindell to file an additional affidavit was also denied as non-responsive, immaterial, and duplicative.
Finally, the district court granted summary judgment for the defendant. The court held that Lindell failed to put into dispute McCaughtry’s facts showing that denying Lindell the magazine related to a legitimate penological interest in security and order, especially because Lindell admitted having been involved in two recent altercations with minority inmates who dis*876agreed with his known supremacist views, and because WCI incarcerates numerous inmates with violent histories.
On appeal, Lindell contests multiple procedural rulings. First, he argues that he should have been permitted to amend his complaint after the court dismissed the case under § 1915A because the defendants had not yet filed a responsive pleading. But after a final judgment, including a judgment under § 1915A, the plaintiff may no longer amend as a matter of right. Harris v. City of Auburn, 27 F.3d 1284, 1287 (7th Cir.1999). Any further amendments under Fed.R.Civ.P. 15(a) are left to the discretion of the district court. The district court found and we agree that justice did not require granting Lindell permission to amend. Although we allow liberal pleading, especially for pro se plaintiffs, the district court may deny a motion to amend for a number of reasons, including undue delay, bad faith or dilatory motive on the part of the movant, repeated failure to cure deficiencies, undue prejudice to the opposing party, and futility. Chavez v. Ill. State Police, 251 F.3d 612, 632 (7th Cir.2001). Lindell’s proposed amendments were either frivolous or duplicative and therefore futile.
Lindell next challenges the district court’s dismissal of the unidentified mail-room employees, claiming that it violated the remand instructions in our earlier decision because that decision did not provide for a dismissal. But the district court had authority to take this action under Fed. R.Civ.P. 21, which provides that “[plarties may be dropped or added by order of the court on motion of any party or of its own initiative at any stage of the action and on such terms as are just.” We agree with the district court that Lindell was not prejudiced by the dismissal because, as we determine here, he could not have won his suit against the Doe defendants.
Lindell charges that the district court should have compelled McCaughtry to identify the Does, sanctioned him and his attorney, and held them in contempt. He claims that it was an abuse of discretion to declare his motions for sanctions moot because a court must state a reason when denying such motions. However, to find a motion moot is to give a reason for denying it. We allow district courts wide latitude in fashioning sanctions and will not disturb their decisions unless they are unreasonable. Research Sys. Corp. v. IP-SOS Publicite, 276 F.3d 914, 920 (7th Cir.2002). Since Lindell has not shown that McCaughtry committed any misconduct, we cannot say that the district court’s decision was unreasonable.
Lindell also challenges the denial of a number of his discovery requests, most importantly, his request for a copy of Issue # 45. He insists that he is entitled to the copy for the purpose of challenging the reasonableness of banning the issue. Since the district court narrowed the questions in the case to the single question whether the rejection of Issue #45 was related to a legitimate penological purpose, most of the discovery requests were simply irrelevant. Issue #45 is relevant; however, we agree with the district court that inmates must not be allowed to evade security restrictions by the simple expedient of filing suit and obtaining prohibited materials through discovery. Our liberal discovery policy does not mean that Lin-dell has an absolute right to everything relevant to his case. District courts have broad discretion in discovery matters. Packman v. Chicago Tribune Co., 267 F.3d 628, 646-47 (7th Cir.2001). We will not disturb a district court’s judgment without a clear showing that the denial of discovery resulted in actual and substantial prejudice to the movant. Id. Because Lindell *877was given a summary of the passages the prison security office found objectionable, we find that he had material to prepare his case.
Next, Lindell objects to the district court’s decision to admit three supplemental proposed facts offered by McCaughtry after he moved for summary judgment and to its decision not to admit an additional proposed affidavit Lindell offered himself. However, we believe these evidentiary rulings to be reasonable, and even if there were error, it would be harmless. One more affidavit of the same sort submitted earlier will not make the difference between winning and losing for Lindell, and the moving party does not need to support its motion for summary judgment with any affidavits or other material negating the opponent’s claim, see Celotex Corp. v. Catrett, 477 U.S. 317, 323, 106 S.Ct. 2548, 91 L.Ed.2d 265 (1986).
We now move to the district court’s grant of summary judgment, which we review de novo. Chavez, 251 F.3d at 635. The standard for determining whether a prison’s censorship of an incoming publication violates an inmate’s constitutional rights is whether the regulation that supports the censorship is “reasonably related to legitimate penological interests.” Thornburgh v. Abbott, 490 U.S. 401, 404, 109 S.Ct. 1874, 104 L.Ed.2d 459 (1989) (quoting Turner v. Safley, 482 U.S. 78, 89, 107 S.Ct. 2254, 96 L.Ed.2d 64 (1987)). We give great deference to prison administrators because we are aware that we sit at some distance from the practical problems of prison management. See id. at 407-08, 109 S.Ct. 1874. However, we are required to measure the reasonableness of a challenged regulation by using the four-factor test given in Turner v. Safley: (1) there must be a rational connection between the regulation and the legitimate governmental interest put forward to justify it, and the governmental interest must be a neutral one; (2) there must be alternative means of exercising the right burdened by the regulation that remain open to inmates; (3) the court must consider the impact accommodation of the asserted right would have on guards and other inmates and on the allocation of prison resources, giving more deference to the judgment of prison administrators where accommodation would have a “ripple effect” on others in the community; (4) finally the absence of ready alternatives is evidence of the reasonableness of a prison regulation. Turner, 482 U.S. at 89-91, 107 S.Ct. 2254.
Lindell’s first argument is that the policy is not related to a legitimate penological interest because racist literature does not cause violence. This is apparently an attack on the rationality of the regulation. In support of his assertion, Lindell submitted to the district court fourteen affidavits from prisoners of different races and religious affiliations. The prisoners stated that they had “no problem” with Lindell possessing racist literature and that other things such as abuse from prison staff were responsible for the violence that occurred in the prison. However, these opinions of Lindell’s fellow inmates are not probative as to whether hateful and violent material of the kind represented in Issue #45 threatens institutional security and order. For one thing, none of these affiants ever saw Issue #45. For another, even if the affiants were completely credible and accurate as to their own sensitivities, which is a big if, they may hardly be considered representative, in number or character, of the inmate population as a whole.
Furthermore, we note that prohibiting violent and hateful literature has been recognized by several of our sister circuits as reasonably related to legitimate penologi*878cal interests. See Chriceol v. Phillips, 169 F.3d 313, 316 (5th Cir.1999) (upholding a state “policy of withholding mail that advocates racial, religious, or national hatred that creates a serious danger of violence”); Murphy v. Missouri Dep’t of Corr., 814 F.2d 1252, 1257 (8th Cir.1987) (state regulation banning racist publication constitutional provided that it is “limited to those materials that advocate violence or that are so racially inflammatory as to be reasonably likely to cause violence at the prison”); McCabe v. Arave, 827 F.2d 634, 638 (9th Cir.1987) (stating that materials that advocate violence or that are so racially inflammatory as to be reasonably likely to cause violence “can certainly be banned”). Lindell is correct in claiming that racist perspectives alone cannot serve as a basis for restriction. It is a violation of the First Amendment for prison officials to censor expression of “inflammatory political, racial, religious or other views” for reasons unrelated to legitimate penological interests. Lindell v. Frank, 377 F.3d 655, 658 (7th Cir.2004). However, when racism becomes violent or inflammatory, courts have not required extrinsic proofs that it is rational to ban it. We said in Shimer v. Washington, 100 F.3d 506, 509-10 (7th Cir.1996), that the “prison administration must proffer some evidence to support its restriction of prison guards’ constitutional rights,” but there we faced a situation where the policy was “not currently obvious” to the court. Our interest was in making sure that prison administrators could not “avoid court scrutiny by reflexive, rote assertions.” See also Lawson v. Singletary, 85 F.3d 502, 503 (11th Cir.1996) (holding that requiring “a causal link between text that [the administration] wants to remove and actual incidents of violence ... constitutes insufficient deference to the judgment of prison authorities”).
We do not have a precise burden standard to apply. The Supreme Court has expressly disclaimed that Thornburgh has any implications for “any question of evidentiary burdens or burden shifting.” However, it did approve the following standard of the district court in that case as being “sufficiently close” to the correct standard “for present purposes:” “the Bureau [must] articulate a relationship between its regulations (and practices) and legitimate penological objectives such as internal security. Once the Bureau meets that requirement, the plaintiffs must show by ‘substantial evidence’ that the defendants have ‘exaggerated their response’ to the problems the regulations address.” Thornburgh, 490 U.S. at 414 n. 12, 109 S.Ct. 1874. In any case, whatever it is, we think McCaughtry has met his burden.
Lindell concedes that there is “hatred” in the publication but considers that this is protected speech. We interpret “hatred” for the purposes of the DOC regulation to mean expressions that are unreasonably virulent, and note that Wis. Admin. Code § DOC 309.05(2)(b)(l) does not simply ban hateful publications, but those that “[t]each or advocate violence or hatred and present a danger to institutional security and order”). It is a judgment call that we second that the expressions in Issue # 45 meet that standard.
Lindell does not argue that there are no alternative means for him to exercise his right to receive communications concerning his supremacist/religious beliefs, so we will not need to consider the second Turner factor. The third factor, concerning impact on other inmates and guards, weighs against Lindell, so we will only note that, according to Thornburgh, “[t]he problem is not ... in the individual reading the materials in most cases. The problem is in the material getting into the prison.” Thornburgh, 490 U.S. at 413, 109 S.Ct. 1874. “Once in the prison, material of this kind reasonably may be expected to *879circulate among prisoners, with the concomitant potential for coordinated disruptive conduct.” Id. at 412, 109 S.Ct. 1874.
Lindell does suggest that the DOC should have redacted the offending portions of the magazine and given him the rest, which invokes the fourth Turner factor. However, we are unwilling to say that the DOC should be required to redact publications when the Supreme Court approved the Federal Bureau of Prisons’ retention of the “all-or-nothing” rule, which bans the entire publication if anything is found that may threaten security and order. Id. at 418-19,109 S.Ct. 1874.
Finally Lindell argues that the application of the regulation is “not neutral” because other “inflammatory” materials are already allowed in the prison. Under Turner and Thornburgh, “neutral” simply means that distinctions between publications are drawn solely on the basis of their potential implications for prison security. We cannot say that McCaughtry is being unreasonable in finding different implications for security between materials such as the Bible and the works of Nietzsche, and Pagan Revival, and we do not think this is inconsistent. Moreover, even if there is inconsistency on the part of the prison administration, it does not necessarily mean that the decisions are unconstitutional. Id. at 417 n. 5, 109 S.Ct. 1874.
Lindell must do more than allege some doubt as to the material facts. A “plaintiffs speculation” is not enough to overcome summary judgment. Packman, 267 F.3d at 637.
AFFIRMED.